The prospect of actually "finishing" physics could be a nightmare for those working in it.
I had a friend who was a brilliant pure mathematician, his teachers and friends agreed he fully deserved to do a PhD in complex analysis - but complex analysis was essentially done in the 19th century, there's just not a lot of research opportunities there.
I have a friend who is a professor with a PhD in complex analysis. Multivariable complex analysis (along with the theories of complex manifolds and analytic spaces) is very far from over.
Heck, one of the Millennium Problems (the Hodge conjecture) is in complex geometry (granted, in the more algebraically flavoured part of it).
I can't really believe that this is the case, although I'm nowhere near being a mathematician. The very nature of knowledge is that every new understanding or discovery raises even more questions.
So complex analysis is pretty much done? Well, could the methods of complex analysis suggest analogical methods in other analytic fields? What work could be done at the intersection of complex analysis and X, when X is any other mathematical field? Also, I hear it is frequently useful in the solution of physical problems. There are many unsolved physical problems that could benefit from being reviewed from a complex analysis perspective.
> There's just not a lot of research opportunities
Maybe this is the crux of the matter; it is not that there is any lack f work still to be done in complex analysis, but there are few research areas in the field that can or are able to attract funding.
I had a friend who was a brilliant pure mathematician, his teachers and friends agreed he fully deserved to do a PhD in complex analysis - but complex analysis was essentially done in the 19th century, there's just not a lot of research opportunities there.