  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.
