I don't know much about this controversy, but looking at the Wikipedia page for the Four Color Theorem, it looks like since the initial Appel-Haken result, mathematicians have continued to work on the Four Color Theorem to try to find a simpler proof. So did they really accept it?
There are a lot of proofs for Pythagoras' Theorem and other theorems.
There are mappings between complex numbers and 2D matrices allowing problems to be solved in either domain.
There's research into The Langlands Program looking to connect number theory and harmonic analysis.
There's research into Category Theory looking to define core concepts and relate them do different fields so that results in one field can be applied to another due to equivalence.
