  Non-Hermitian Wishart matrices were introduced in the context of quantum
chromodynamics with a baryon chemical potential. These provide chiral
extensions of the elliptic Ginibre ensembles as well as non-Hermitian
extensions of the classical Wishart/Laguerre ensembles. In this work, we
investigate eigenvalues of non-Hermitian Wishart matrices in the symmetry
classes of complex and symplectic Ginibre ensembles. We introduce a generalised
Christoffel-Darboux formula in the form of a certain second-order differential
equation, offering a unified and robust method for analyzing correlation
functions across all scaling regimes in the model. By employing this method, we
derive universal bulk and edge scaling limits for eigenvalue correlations at
both strong and weak non-Hermiticity.
