  By using the method of orthogonal polynomials we analyze the statistical
properties of complex eigenvalues of random matrices describing a crossover
from Hermitian matrices characterized by the Wigner- Dyson statistics of real
eigenvalues to strongly non-Hermitian ones whose complex eigenvalues were
studied by Ginibre.
  Two-point statistical measures (as e.g. spectral form factor, number variance
and small distance behavior of the nearest neighbor distance distribution
$p(s)$) are studied in more detail. In particular, we found that the latter
function may exhibit unusual behavior $p(s)\propto s^{5/2}$ for some parameter
values.
