  We address the question of how the celebrated universality of local
correlations for the real eigenvalues of Hermitian random matrices of size NxN
can be extended to complex eigenvalues in the case of random matrices without
symmetry. Depending on the location in the spectrum, particular large-N limits
(the so-called weakly non-Hermitian limits) lead to one-parameter deformations
of the Airy, sine and Bessel kernels into the complex plane. This makes their
universality highly suggestive for all symmetry classes. We compare all the
known limiting real kernels and their deformations into the complex plane for
all three Dyson indices beta=1,2,4, corresponding to real, complex and
quaternion real matrix elements. This includes new results for Airy kernels in
the complex plane for beta=1,4. For the Gaussian ensembles of elliptic Ginibre
and non-Hermitian Wishart matrices we give all kernels for finite N, built from
orthogonal and skew-orthogonal polynomials in the complex plane. Finally we
comment on how much is known to date regarding the universality of these
kernels in the complex plane, and discuss some open problems.
