Complex matrix to Block Matrix

Gaso
How can I redefine my Complex matrix to a Block matrix, similar as matrix representation of complex number.
I need a Real an Imaginary part as real numbers, to find eigenvalues and eigenvectors with my solver, which works only with Real matrices.

My matrix element is:
$$M_{mn}=\int^{tk}_{0} -i Exp[i E_{mn} t] V_{mn}(t) dt$$
The integrals are easy to write, but are complex.
I want to write matrix M as new matrix of dimension 2N to find eigenvectors and eigenvalues of that matrix, which are also complex.