Putting probability distributions over observer moments is a critical first step in this program.
There are some problems with the notion of 'observer moments'. I'm inclined to think they are unresolvable, but perhaps you have some ideas for how to tackle them.
I've already mentioned the problem of the 'boundary' of a subjective state. For instance, consider an old memory which it would take you a long time to 'dredge up'. How is that different in principle from having some information written down somewhere in a notebook in front of you? Is that old memory part of your 'observer moment'? (But then how about a slightly fresher memory)? Is the notebook? (But then how about that thick reference book on your shelf)? It seems obvious to me that there's no principled line to be drawn here.
Then there's the problem of whether a given system is an observer or not. For instance, Dennett is notorious for (correctly!) attributing 'intentional states' to a thermostat: you can view it as an agent who believes the temperature is x and wants it to be y. But is a thermostat an observer? Presumably not, but again it seems that there's no principled line to be drawn between thermostats and people.
And then there's the problem of 'how many observers'. E.g. Is a split-brain patient two observers or one? How about an ordinary person? How much of the corpus callosum needs to be severed to get two observers?
Finally, if A is much much cleverer, more alert, and knowledgeable than B then A ought to have a greater density of 'observer moments' than B? But exactly how much greater? The idea that there's a principled way of determining this seems overoptimistic.
(Going a little off topic: I've actually been thinking along vaguely similar lines to you just lately. I've been trying to devise an approach to the mind-body problem in terms of "perspectives". My overall goal was to try to 'do justice to' the common intuition that a subjective point of view must be determinate and cannot only half-exist, while maintaining the substance of Dennett's position which implies that there may be no fact of the matter as to whether a person is conscious and what they're conscious of. My key idea is that (a) there exist genuine facts of the form "From perspective P, such-and-such would be consciously experienced" but (b) the perspectives themselves do not exist - they're not part of the "state of the world". (Instead, they're "perspectives from which the state of the world manifests itself"). An analogy would be how, in mathematical logic, we have this duality between theories and models, or more broadly, between syntax and semantics. The "syntax side" enables you to state and prove things about stuff that exists, but the syntax itself doesn't exist. (In the same way that there's no such set as ZFC.)
My 'perspectives' are essentially the same as your 'pointers-to-observers'. However, I'd want to stress the fact that perspectives are ubiquitous - e.g. you can take the perspective of a rock, or a thermostat. And you can take the perspective of a person in many different ways, with no fact of the matter about which is right, or even whether it's right to take one. (But from any given perspective, all the subjective facts are nice and 'determinate'.)
It never occurred to me to try to consider the Kolmogorov complexities of perspectives. It's an interesting idea, but it's hard to wrap one's head around given that there are an unlimited number of ways of defining the same person's perspective.)
I believe that the execution of a certain computation is a necessary and sufficient condition for my conscious experience. Following Tegmark, by "execution" I don't refer to any notion of physical existence--I suspect that the mathematical possibility of my thoughts implies conscious experience. By observing the world and postulating my own representativeness, I conjecture the following measure on different possible experiences: the probability of any particular experience drops off exponentially with the complexity required to specify the corresponding computation.
It is typical to use some complexity prior to select a universe, and then to appeal to some different notion to handle the remaining anthropic reasoning (to ask: how many beings have my experiences within this universe?). What I am suggesting is to instead apply a complexity prior to our experiences directly.
If I believe a brain embodying my thoughts exists in some simple universe, then my thoughts can be described precisely by first describing that universe and then pointing to the network of causal relationships which constitute my thoughts. If I have seen enough of the universe, then this will be the most concise description consistent with my experiences. If there are many "copies" of that brain within the universe, then it becomes that much easier to specify my thoughts. In fact, it is easy to check that you recover essentially intuitive anthropics in this way.
This prior has a significant impact on the status of simulations. In general, making two simulations of a brain puts twice as much probability on the associated experiences. However, we no longer maintain substrate independence (which I now consider a good thing, having discovered that my naive treatment of anthropics for simulations is wildly inconsistent). The significance of a particular simulation depends on how difficult it is to specify (within the simple universe containing that simulation) the causal relationships that represent its thoughts. So if we imagine the process of "splitting" a simulation running on a computer which is two atoms thick, we predict that (at least under certain circumstances) the number of copies doubles but the complexity of specifying each one increases to cancel the effect.
This prior also gives precise answers to anthropic questions in cosmology. Even in an infinite universe, description complexity still answers questions such as "how much of you is there? Why aren't you a Boltzmann brain?" (of course this still supposes that a complexity prior is applicable to the universe).
This prior also, at least in principle, tells you how to handle anthropics across quantum worlds. Either it can account for the Born probabilities (possibly in conjunction with some additional physics, like stray probability mass wandering in from nearby incoherent worlds) or it can't. In that sense, this theory makes a testable "prediction." If it does correctly explain the Born probabilities, then I feel significantly more confidence in my understanding of quantum mechanics and in this version of a mathematical multiverse. If it doesn't, then I tentatively reject this version of a mathematical multiverse (tentatively because there could certainly be more complicated things still happening in quantum mechanics, and I don't yet know of any satisfactory explanation for the Born probabilities).
Edit: this idea is exactly the same as UDASSA as initially articulated by Wei Dai. I think it is a shame that the arguments aren't more widespread, since it very cleanly resolves some of my confusion about simulations and infinite cosmologies. My only contribution appears to be a slightly more concrete plan for calculating (or failing to calculate) the Born probabilities; I will report back later about how the computation goes.