Can every non-hermitian matrix be written as a sum of hermitan matrices? If $X$ is non-hermitian i.e., $X^\dagger = (X^*)^T \ne X$, can one express it as $X = \sum_i \alpha_i Y_i$ where  $\alpha_i$ is a complex scalar and $Y_i$ is hermitian?

If you really do intend to take complex-linear combinations, then, yes, this is possible:
$$
X \;=\; {1\over 2}\cdot (X+X^*) + {i\over 2}\cdot \Big({X-X^*\over i}\Big)
$$
and both expressions in parentheses are hermitian.
