Why a complex symmetric matrix is not diagonalizible?

As Chris Godsil and Dietrich Burde pointed out it's because $\langle x,y \rangle =x^*y=0$ which is the orthogonality condition on complex vectors does not imply that $x^Ty=0$ which is the complex symmetry condition.

So the Gram Schmidt process actually will produce orthogonal vectors, but they will not be able to diagonalize the matrix.

First of all, there is an easy counterexample. The complex symmetric matrix $$\begin{pmatrix} 1 & i \\ i & -1 \end{pmatrix}$$ is not diagonalizable, because trace and determinant are zero, but the matrix is not zero. Now try the Gram-Schmidt process in this example.