  We study the distribution of eigenvalues of almost-Hermitian random matrices
associated with the classical Gaussian and Laguerre unitary ensembles. In the
almost-Hermitian setting, which was pioneered by Fyodorov, Khoruzhenko and
Sommers in the case of GUE, the eigenvalues are not confined to the real axis,
but instead have imaginary parts which vary within a narrow ``band'' about the
real line, of height proportional to $\tfrac 1 N$, where $N$ denotes the size
of the matrices. We study vertical cross-sections of the 1-point density as
well as microscopic scaling limits, and we compare with other results which
have appeared in the literature in recent years. Our approach uses Ward's
equation and a property which we call ``cross-section convergence'', which
relates the large-$N$ limit of the cross-sections of the density of eigenvalues
with the equilibrium density for the corresponding Hermitian ensemble: the
semi-circle law for GUE and the Marchenko-Pastur law for LUE. As an application
of our approach, we prove the bulk universality of the almost-circular
ensembles.
