what's wrong with this argument?

So i need help finding what is wrong with this argument:

Notice A= $\displaystyle (A-A*)/2+(A+A*)/2$

and both $\displaystyle (A-A*)/2 and (A+A*)/2$ 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 $\displaystyle (A-A*)/2+(A+A*)/2$ 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)