No, not every matrix over C is diagonalizable. Indeed, the standard example (0100) remains non-diagonalizable over the complex numbers.

How do you know if a matrix is diagonalizable over C?

Let A be an n × n matrix with complex entries. Then it has at least one complex eigenvalue. It has exactly n complex eigenvalues if each eigenvalue is counted corresponding to its (algebraic) multiplicity. If the characteristic polynomial of A has n distinct linear factors then A is diagonalizable over C.

Are all matrices diagonalizable over C?

Every matrix is not diagonalisable. Take for example non-zero nilpotent matrices.