  A non-Hermitean extension of paradigmatic Wishart random matrices is
introduced to set up a theoretical framework for statistical analysis of (real,
complex and real quaternion) stochastic time series representing two "remote"
complex systems. The first paper in a series provides a detailed spectral
theory of non-Hermitean Wishart random matrices composed of complex valued
entries. The great emphasis is placed on an asymptotic analysis of the mean
eigenvalue density for which we derive, among other results, a complex-plane
analogue of the Marchenko-Pastur law. A surprising connection with a class of
matrix models previously invented in the context of quantum chromodynamics is
pointed out.
