You have answered your own question. The basis that diagonalises (A–A*)/2 need not be the same as the basis that diagonalises (A+A*)/2. So there is no reason why there should be a single basis that diagonalises both.
So i need help finding what is wrong with this argument:
Notice A=
and both are diagonalizable (using the complex spectral theorem)
thus in some basis A will be the sum of two diagonal matrices and hence diagonal.
Therefore A is diagonalizable.
Is the thing that is wrong with this argument the fact that would both need to be diagonalized using the SAME basis in order for their sum to make a diagonal matrix representative of A?
(A*= the conjugate transpose of A)