The completeness of physics refers, not to the confidence that all the problems of physics will be solved or that it will be able to explain everything coming under the heading of the “really real”. But it’s the view that nothing in the domain of the really real falls outside of the domain of the really physical. Reality is not composed to two radically different kinds of stuff. But of one kind only, and that is physical stuff. This is the view that physics is complete.
Metaphysically speaking this position is called ontological monism. There is but one kind of furniture in the universe. For the ontological dualist there are two kinds. For the ontological dualist to be correct, physics would be incomplete in that the laws and principles of physics would leave unaccounted for this other immaterial, nonphysical aspect of the really existant.
The ontological monist doesn’t need to forbid the use of mentalistic terms or concepts. These can be kept for folk psychology. Long after we discover that lightning is an electrical discharge, we speak of lightning as we still do. To use the phrase advanced by Donald Davidson, we may well have to settle for an “anomalous monism”, that is, a recognition of the validity of ontological monism, but also the recognition of that our mentalistic terms and jargon will never be reduced to physical events and processes. Still, that there are social and discursive justifications for using a certain set of terms does not establish an ontological justification for assuming the reality of what such terms refer to. There may be good folk psychological reasons for referring to Lady Luck, but this is distinct from the question of whether either exists.