  The elliptic Ginibre ensemble of complex non-Hermitian random matrices allows
to interpolate between the rotational invariant Ginibre ensemble and the
Gaussian unitary ensemble of Hermitian random matrices. It corresponds to a
two-dimensional one-component Coulomb gas in a quadrupolar field, at inverse
temperature $\beta=2$. Furthermore, it represents a determinantal point process
in the complex plane with corresponding kernel of planar Hermite polynomials.
Our main tool is a saddle point analysis of a single contour integral
representation of this kernel. We provide a unifying approach to rigorously
derive several known and new results of local and global spectral statistics,
including in higher dimensions. First, we prove the global statistics in the
elliptic Ginibre ensemble first derived by Forrester and Jancovici. The
limiting kernel receives its main contribution from the boundary of the
limiting elliptic droplet of support. In the Hermitian limit, there is a know
correspondence between non-interacting fermions in a trap in $d$ real
dimensions $\mathbb{R}^d$ and the $d$-dimensional harmonic oscillator. We
present a rigorous proof for the local $d$-dimensional bulk (sine-) and edge
(Airy-) kernel first defined by Dean et al., complementing recent results by
Deleporte and Lambert. Using the same relation to the $d$-dimensional harmonic
oscillator in $d$ complex dimensions $\mathbb{C}^d$, we provide new local bulk
and edge statistics at weak and strong non-Hermiticity, where the former
interpolates between correlations in $d$ real and $d$ complex dimensions. For
$\mathbb{C}^d$ with $d=1$ this corresponds to non-interacting fermions in a
rotating trap.
