There's this longstanding observation that we live in a "fine-tuned universe," which is to say that several fundamental physical constants (e.g. strength of gravity, electromagnetism) appear to be set perfectly for us. If they were any different, even by one part in a billion, the universe would have evolved very differently and failed to give rise to sufficient complexity to eventually enable life as we know it. Which, at first glance, seems awfully suspicious.
One proposed explanation for this is the Anthropic Principle, which tries to couch the whole thing as a selection effect. The claim is that obviously the cosmological constants had to be set that way, because if they weren't, nobody would be around to notice, and therefore no meaningful conclusions can be drawn. While the basic premise makes sense, arguing that there's nothing to be inferred strikes me as a load of horseshit.
If there were only one universe (ours), with one set of laws and constants, the probability of it just happening to work out like this is so close to 0 as to be negligible, barring the possibility that it was deliberately engineered that way (which is a whole different discussion.) So, it seems extremely likely to me that this should be taken as strong evidence in support of a multiverse of one form or another, consisting of a vast number (or more likely, an infinite number) of universes with varying laws of physics and initial conditions. This would neatly account for our seemingly extraordinarily unlikely circumstances, and incidentally explains and recolors Occam's Razor not just as a heuristic, but a de facto selection pressure in its own right.
Well, now let's look at consciousness and experience as computation. If you discount the notion of a soul, or other hand-wavy quantum mechanical effects, all we are is our own brain, fed appropriate input. The brain is presumably Turing-complete, essentially a computer— or more to the point, operates on principles that could be precisely modeled on a computer. You could be running on a PC right now, and if the model and coding were all correct, there'd be no subjective difference for you.
Of course, that's just The Matrix 101. But the problem is, things get weirder. It turns out that a huge variety of systems can be made to be Turing-complete, which is to say capable of carrying out any computation that a PC (or a brain) could execute. Bored CS students have designed Turing machines out of tinkertoys and Minecraft levels— hell, you could even have a hundred Tibetan monks pushing around beads on abacuses, and so long as they're conducting computation that can be isomorphically mapped bijectively to neural wetware, the end result is that you, your whole life and apparent consciousness and free will could be no more than a consequence of those beads getting pushed. Although this sounds a little out there, it all logically follows, and I think a growing consensus is emerging about it among people working in these fields.
So what's the first thing got to do with the second?
If we assume all possible universes exist and the level of algorithmic complexity of each universe is uniformly distributed over that infinite set (which seems like a plausible assumption), then universes dominated by the simplest of laws will be infinitely dense relative to more rarefied and complexity-heavy universes like our own. I expect that many of those universes will be appropriately configured such that they end up running a huge amount of computation, whether through a physical substrate (think an infinitely physically large universe consisting of random quantum fluctuations affecting clumps of molecules that tend to form NAND gates) or directly in some kind of even more abstract mathematical formalism.
If that's true, then every person that does exist or could exist and every life they could live would be inevitably simulated by just a single one of these Turing-verses. Which would mean, in turn, that it is essentially mathematically certain that you're not really here in the sense that you think, but are instead one of those simulations in a completely different and alien universe.
Now the question is how one could falsify this whole hypothesis, or better yet, if it turns out to be true, how one could arrange things in our universe such that we tunnel our way out into some control of the underlying Turing-verse.
And it's also conceivable that carrying out computation itself is not necessary to give rise to our subjective reality. It may be that just the pattern itself, in static form, is sufficient. That would be a beautiful thing, because it would open up the possibility that the only thing that exists in all of creation, across all universes and all time, is a single number encoding all things, in exactly the same way that any phone number, or book, or data representations of entire lives and worlds are contained within the digits of the number pi.