  We solve the complex extension of the chiral Gaussian Symplectic Ensemble,
defined as a Gaussian two-matrix model of chiral non-Hermitian quaternion real
matrices. This leads to the appearance of Laguerre polynomials in the complex
plane and we prove their orthogonality. Alternatively, a complex eigenvalue
representation of this ensemble is given for general weight functions. All
k-point correlation functions of complex eigenvalues are given in terms of the
corresponding skew orthogonal polynomials in the complex plane for finite-N,
where N is the matrix size or number of eigenvalues, respectively. We also
allow for an arbitrary number of complex conjugate pairs of characteristic
polynomials in the weight function, corresponding to massive quark flavours in
applications to field theory. Explicit expressions are given in the large-N
limit at both weak and strong non-Hermiticity for the weight of the Gaussian
two-matrix model. This model can be mapped to the complex Dirac operator
spectrum with non-vanishing chemical potential. It belongs to the symmetry
class of either the adjoint representation or two colours in the fundamental
representation using staggered lattice fermions.
