With enough motivation, you can learn new areas and go there. I started my first two papers in complex analysis, related to the Schrodinger equation. I am now doing algebraic combinatorics related to representation theory, some quasi-symmetric functions and enumerative aspects of combinatorics. On the side, I have also worked a bit on polytopes, and an unfinished project which are related to invariant measures and Julia fractals.

There is no point of taking a second PhD - having a PhD means you should be mature enough to read research articles and study mathematics by yourself without having to take classes. You are also (I hope) familiar with the ethical aspects of research and the submission/review process. You should also know what constitutes a well-written paper and you are now familiar with LaTeX and mathematical software and how to present mathematics in form of posters and talks.