The Universe of Doctor Who is an odd one to consider within the context of the theological theory of Pandeism -- after all, Pandeism is logic-based, whereas the Whovian realm is perhaps somewhat more.... whimsical. A bit silly, even, not only in the behavior of its denizens, but in the operations of time and space and physics.
For the uninitiated, the world of Doctor Who centers on the adventures of the Doctor -- a Time Lord from the once-great (and now sadly destroyed) planet Gallifrey. The Doctor inhabits a time-and-space ship of sorts called the TARDIS, which externally looks like a 1960's British police box, but is enormous on the inside, and which enables him to travel to any point in time and space. This he does, fighting a variety of enemies, old recurring ones as often as new ones, and generally (though not always entirely) saving the day. In so doing, the Doctor has racked up several dozen (human) lifetimes worth of positively undreamt of experiences. The theological theory of Pandeism, on the other hand, centers on the idea of a Creator which, instead of setting forth a Universe separate from itself, wholly becomes our Universe (or, perhaps, the Doctor's Universe) so as to learn from the myriad combination of experiences of those beings which come about in such a Universe.
As with all fictional worlds, there are several directions from which the particulars of this world allow for the pandeistic enquiry. One is in terms of what the occupants of this world believe. But belief is actually not so important (although, perhaps what the Doctor believes is more important, as we shall consider). Another is the actuality of that Universe, its physics and governing dynamics, including the actual presence (or absence) of supernatural phenomena.
What is believed
Like many science fiction universes, that of Doctor Who is well-stocked with religiosity. And, like many science fiction universes, religious belief is often shown to be errant and manipulable. Things deemed by common folk as supernatural are routinely shown by the Doctor as having a proper scientific explanation -- typically in the form of aliens pretending to be gods (or being mistaken for them; or, indeed, the Doctor himself being mistaken for one). Except that those scientific explanations are themselves quite often things which defy scientific reason. A very recent example is the episode, "The Rings of Akhaten." Here, the Doctor and his companion, Clara, visit a planetary system where instead of money, things are bartered for with items having 'psychometric' value, being the source of some emotional attachment. The denizens of this world are sending a young girl to join in an eternal chorus being sung to an especially demanding god--one which might awaken and destroy the girl if it is not placated with her singing. But something goes wrong with the song and the god awakens and threatens to devour the entire system, and then others beyond it. Fortunately, after the Doctor fails to placate it with his own life story, Clara is able to overcome this god by feeding it "the most important leaf in human history," one which allowed her parents to meet, but which now represents the infinite possibilities lost with her mother's untimely death.
And so, instead of an actual 'god,' we find in this episode that the culprit was simply a run-of-the-mill unfathomably powerful living planet which eats sentimental attachments, and can be stopped by a single leaf which happens to be imbued with infinite possibilities. Does that not sound like magic? Beneath the veneer of this being a commonsense scientific explanation, does that not smack of a wholly supernatural account springing straight from ancient mythology? But in the world or the Doctor, it is indeed routine -- one supernatural explanation explained away by another one framed as scientific. But in a pandeistic world the explanation would more likely actually be unequivocally and testably scientific, and not an equally implausible device simply dressed in scientific language.
What the Doctor believes
The Doctor's own views would be of especially remarkable weight in the Doctor's Universe because he clearly possess vastly more knowledge then the typical human can even grasp (even if he does often act as if he were confused by human romantic and sexual relationships). In "The Parting of the Ways," companion Rose Tyler has looked into the heart of the Tardis and so was being overwhelmed by having obtained the ability to see "everything. All that is, all that was, all that ever could be." The Doctor sympathetically responds, "That's what I see. All the time. Doesn't it drive you mad?"
With all of this knowledge and centuries of wisdom, it is notable that the Doctor never prays to any deity. He never endorses the truth of any scripture or doctrine, invariably immediately doubting any reliance upon supernatural explanations (and for good reason, based on his experience in the matter). But neither does the Doctor ever clearly express outright atheism.
Such is the case even when the Doctor confronts a creature claiming to be the Devil itself, in "The Satan Pit." Here, the Doctor is visiting an expedition to an 'impossible planet' orbiting a black hole. The planet turns out to be a prison for a powerful evil, one which awakens and claims to be that universal bogeyman of badness, Satan. The Doctor asks, "If you are the Beast, then answer me this: Which one, hmm? Because the universe has been busy since you’ve been gone. There's more religions than there are planets in the sky. There's the Arkaphets, Christianity, Pash-Pash, New Judaism, San Claar, Church of the Tin Vagabond. Which devil are you?" Naturally, the creature claims to be "All of them," and when the Doctor asks when was it imprisoned here, it answers "Before time and light and space and matter. Before the cataclysm. Before this universe was created." This, the Doctor doubts: "You can't have come from before the universe. That's impossible." "Is that your religion?" the creature asks. "It's a belief," the Doctor replies.
Later, the Doctor explains, "If that thing had said it was from beyond the universe, I'd have believed it. But before? Impossible." On closer examination, the creature's claim is indeed impractical -- it's confinement seems to hinge on technology and principles of physics which necessarily postdate the creation of our Universe. And naturally, it could not have engaged in the evils attributed to supernatural bogeymen had it been incapacitated for all those billions of years. But as to the Doctor's belief that nothing could come from before our Universe, perhaps this extends from the understanding in modern physics of time itself as a function of our Universe, with "before the Universe" itself being akin to "North of the North Pole." But if time is a function of our Universe, it would seem to follow that "beyond our Universe" is the same as "beyond time," and so the Doctor's understanding would not preclude an entity from existing outside the bounds of time itself.
What people believe about the Doctor
One theory popular amongst some fans (though derided by most) is that the Doctor is "God." (PBS even dares ask, Is Doctor Who a Religion?, though not on this point especially). Examples abound of the sort of thing which fuels this Doctor-as-deity sort of thinking. One can be found in the episode, New Earth, when the Doctor makes a life-or-death call an anthropomorphic cat nun asks, "And who are you to make this decision?" The Doctor responds, "I'm the Doctor. And if you don't like it, if you want to take it to a higher authority, then there isn't one. It stops with me."
At the end of "The Wedding of River Song," it is revealed that the Silence, a religious order, have been seeking to prevent the Doctor from answering the question "Doctor who?", believing that "silence will fall when the question is asked". According to Dorium Maldovar, the question was told in this manner: "On the Fields of Trenzalore, at the fall of the Eleventh, when no creature can speak falsely or fail to answer, a question will be asked — a question that must never be answered. The first question, the question that must never be answered, hidden in plain sight, the question that you've been running from all your life. Doctor who?" And why would this be "the first question"? Well, it may mean that the Doctor is God, or it may mean that there is no God, in which case the Doctor is surely about the closest thing to one. Or it may mean that whatever god might exist is not one to which an appeal can be taken, and one for which that would be an appropriate "first question"-- which Pandeism would be an example of, such deity having become the unanswering Universe itself.
And what is
More important than what is simply believed is what we can see of how this Universe functions. After all, people will believe all sorts of wrong things for all sorts of reasons (and, even, will come to believe right things for wrong reasons); but are there clues to its workings which coincide with a the operation of a pandeistic theological model? The Universe of Doctor Who is quite a curious place. Physics seems to function more or less coherently -- gravity and electromagnetic reactions in all the right places. But it is, naturally, a place where time travel is possible. And, interferences with the time stream seem to have varying effects on reality -- pasts and futures change, but the same people seem to show up regardless. The degree to which change occurs is inconstant, and this inconstancy is handwaved away with references to temporal polarities or the like. This is a place where things like teleportation and telepathic communication and mind-occupation can be found (in The Lodger, the Doctor demonstrates the ability to convey his own memories and experiences to another person through a head bonk).
The Universe with which the Doctor is familiar teems with alien species of most every imaginable configuration, from many different hues of humanoids to the country-sized Space Whale of "The Beast Below." These aliens equally range in disposition from friendly to murderous, with species such as the Sontarans who are dedicated to warmaking, and the dreaded Daleks, which worship hate and yearn to wipe out all life in the Universe. These beings differ from life on Earth not simply in appearance, but in a host of more arcane capacities. The Akhaten planet-being is one example. Some beings do indeed exist as nothing more than clouds of energy. Another, closer to home, is the Weeping Angels, creatures which look like old-timey grotesque statues of people, but which eat temporal energy (forcing their victims back in time by a few decades when feeding, and which are incapable of moving when anybody is looking at them). Oddly, the Doctor himself describes the Weeping Angels as a product of evolution, though what sort of evolutionary path could produce such things is well nigh unfathomable. The Daleks and another regular foe, the Cybermen, seem to be examples of transhumanism gone wrong, with efforts to augment living things technologically yielding monsters instead of marvels.
And then there is that quite strange creature, the Doctor. Like all Time Lords, he is effectively immortal -- at least up until he gets killed by temporal radiation or alien spiders or the like enough times to use up all of his "regenerations," and so is no longer able to return with a new actor- er- face. If the exploits of other Time Lords are instructive, he has superhuman physical strength. Despite his immense arsenal of abilities, the Doctor is certainly not, by appearances, infallible or omniscient or omnipotent. In "The Waters of Mars" he attempts to prevent a fixed point in history from occurring, and fails tragically, and with great lamentation of this failure. In "The Angels Take Manhattan," he grieves for his inability to ever see companions Amy and Rory again, because his paradoxical doings with New York's time stream have made it too delicate to withstand any further intervention.
Additionally, as noted before, the Doctor is not the only Time Lord to have graced the Universe (though it seems he is now, having destroyed the rest to save the Universe from a fate worse than death). Once, there was a whole planet of them, presumably all at least possessed of the relative immortality which the Doctor enjoys. True, the Doctor seems special even amongst his fellow Time Lords -- on his trips to Gallifrey in earlier incarnations, he usually got the better of them in when they were at cross purposes. Even the Doctor's most able foe, the Master, was ultimately always defeated, even though his apparent final destruction, in "The End of Time," was essentially a self-sacrificing suicide. The event which this suicidal sacrifice prevented was the attempt of the time-locked Gallifreyans to end the Universe itself, and become beings of pure thought. This is interesting on a number of levels firstly because it unquestioningly assumes that the Gallifreyans do indeed have the ability to end the entire Universe, as well as the ability to initiate and then persist in an existence as beings of pure thought. One must wonder if this is a one-way transition -- if material beings can become thought-beings, does it not follow that thought-beings (or, perhaps' one great thought-being) can become material?
And, if the Doctor is God, then aren't those amongst his fellow Time Lords who can from time to time defeat him?
But, naturally, if Pandeism is true then, well, so are we all. And herein arises a uniquely pandeistic possibility, for perhaps the Doctor knows this (or the possibility of it), and so is simply somebody who happens to have a better handle on the power which accompanies such status (or, indeed, comes from a race of beings so empowered). Time Lords apparently gain some of their depth of knowledge, and perhaps other abilities, from a childhood ritual in which they are made to stare into an untempered schism in the time vortex, which would be a handy metaphor for looking straight into the power of a Universe-creating entity underlying all things. And, interestingly, in the episode, "The Pandorica Opens," River Song laments "I hate good wizards in fairy tales. They always turn out to be him." Would it be any surprise whatsoever if, in this reality, Jesus turned out to be the Doctor in an earlier adventure? Or indeed if Moses or Arjuna or Odysseus or Lao Tze or any other mythohistoric figure turned out to be the selfsame sojourner?