  Non-Hermitian random matrices with symplectic symmetry provide examples for
Pfaffian point processes in the complex plane. These point processes are
characterised by a matrix valued kernel of skew-orthogonal polynomials. We
develop their theory in providing an explicit construction of skew-orthogonal
polynomials in terms of orthogonal polynomials that satisfy a three-term
recurrence relation, for general weight functions in the complex plane. New
examples for symplectic ensembles are provided, based on recent developments in
orthogonal polynomials on planar domains or curves in the complex plane.
Furthermore, Bergman-like kernels of skew-orthogonal Hermite and Laguerre
polynomials are derived, from which the conjectured universality of the
elliptic symplectic Ginibre ensemble and its chiral partner follow in the limit
of strong non-Hermiticity at the origin. A Christoffel perturbation of
skew-orthogonal polynomials as it appears in applications to quantum field
theory is provided.
