  We consider the eigenvalues of symplectic elliptic Ginibre matrices which are
known to form a Pfaffian point process whose correlation kernel can be
expressed in terms of the skew-orthogonal Hermite polynomials. We derive the
scaling limits and the convergence rates of the correlation functions at the
real bulk/edge of the spectrum, which in particular establishes the local
universality at strong non-Hermiticity. Furthermore, we obtain the subleading
corrections of the edge correlation kernels, which depend on the
non-Hermiticity parameter contrary to the universal leading term. Our proofs
are based on the asymptotic behaviour of the complex elliptic Ginibre ensemble
due to Lee and Riser as well as on a version of the Christoffel-Darboux
identity, a differential equation satisfied by the skew-orthogonal polynomial
kernel.
