  In this paper we propose a new method for studying spectral properties of the
non-hermitian random matrix ensembles. Alike complex Green's function encodes,
via discontinuities, the real spectrum of the hermitian ensembles, the proposed
here quaternion extension of the Green's function leads directly to complex
spectrum in case of non-hermitian ensembles and encodes additionally some
spectral properties of the eigenvectors. The standard two-by-two matrix
representation of the quaternions leads to generalization of so-called
matrix-valued resolvent, proposed recently in the context of diagrammatic
methods [1-6]. We argue that quaternion Green's function obeys Free Variables
Calculus [7,8]. In particular, the quaternion functional inverse of the matrix
Green's function, called after [9] Blue's function obeys simple addition law,
as observed some time ago [1,3]. Using this law we derive new, general,
algorithmic and efficient method to find the non-holomorphic Green's function
for all non-hermitian ensembles of the form H+iH', where ensembles H and H' are
independent (free in the sense of Voiculescu [7]) hermitian ensembles from
arbitrary measure. We demonstrate the power of the method by a straightforward
rederivation of spectral properties for several examples of non-hermitian
random matrix models.
