I'd like to "model" the absolute complement of a set $X$ as the ordinal-indexed sequence $\alpha \mapsto V_\alpha \setminus X$ where $V_\alpha$ is the $\alpha$ stage of the cumulative hierarchy. My understanding is that ZFC doesn't support ordinal-indexed sequences, so my question is, what is a good set theory in which to study this concept?
Note that the class-length sequence $V_\alpha$ is definable, so given a set $X$, so is the sequence $V_\alpha\setminus X$ going to be definable (via the parameter $X$).
If you intend to use more, perhaps a theory like ZFC+Global choice; or NBG which is more suitable for handling proper classes.
If you are going to talk about collections of classes, then perhaps it is easier to assume an inaccessible cardinal exists, and have two levels of universes: $V_\kappa$ as the world of sets, and its classes are also sets in the larger universe, allowing you to talk about "all complements" or so, if you'd like.