  We expand on our results in arXiv:2111.02429 to present a broad new class of
gravitational observables in asymptotically Anti-de Sitter space living on
general codimension-zero regions of the bulk spacetime. By taking distinct
limits, these observables can reduce to well-studied holographic complexity
proposals, e.g., the volume of the maximal slice and the action or spacetime
volume of the Wheeler-DeWitt patch. As with the codimension-one family found in
arXiv:2111.02429, these new observables display two key universal features for
the thermofield double state: they grow linearly in time at late times and
reproduce the switchback effect. Hence we argue that any member of this new
class of observables is an equally viable candidate as a gravitational dual of
complexity. Moreover, using the Peierls construction, we show that variations
of the codimension-zero and codimension-one observables are encoded in the
gravitational symplectic form on the semi-classical phase-space, which can then
be mapped to the CFT.
