The interpreter, if it would exist, would have complexity. The useless unconnected calculation in the waterfall/rock, which could be but isn’t usually interpreted, also has complexity.
Your/Aaronson’s claim is that only the fully connected, sensibly interacting calculation matters. I agree that this calculation is important—it’s the only type we should probably consider from a moral standpoint, for example. And the complexity of that calculation certainly seems to be located in the interpreter, not in the rock/waterfall.
But in order to claim that only the externally connected calculation has conscious experience, we would need to have it be the case that these connections are essential to the internal conscious experience even in the “normal” case—and that to me is a strange claim! I find it more natural to assume that there are many internal experiences, but only some interact with the world in a sensible way.
You can also disambiguate between
a) computation that actually interacts in a comprehensible way with the real world and
b) computation that has the same internal structure at least momentarily but doesn’t interact meaningfully with the real world.
I expect that (a) can usually be uniquely pinned down to a specific computation (probably in both senses (1) and (2)), while (b) can’t.
But I also think it’s possible that the interactions, while important for establishing the disambiguated computation that we interact with, are not actually crucial to internal experience, so that the multiple possible computations of type (b) may also be associated with internal experiences—similar to Boltzmann brains.
(I think I got this idea from “Good and Real” by Gary L. Drescher. See sections “2.3 The Problematic Arbitrariness of Representation” and “7.2.3 Consciousness and Subjunctive Reciprocity”)