  We compare the Ornstein-Uhlenbeck process for the Gaussian Unitary Ensemble
to its non-hermitian counterpart - for the complex Ginibre ensemble. We exploit
the mathematical framework based on the generalized Green's functions, which
involves a new, hidden complex variable, in comparison to the standard
treatment of the resolvents. This new variable turns out to be crucial to
understand the pattern of the evolution of non-hermitian systems. The new
feature is the emergence of the coupling between the flow of eigenvalues and
that of left/right eigenvectors. We analyze local and global equilibria for
both systems. Finally, we highlight some unexpected links between both
ensembles.
