  Recently Burkhardt et. al. introduced the $k$-checkerboard random matrix
ensembles, which have a split limiting behavior of the eigenvalues (in the
limit all but $k$ of the eigenvalues are on the order of $\sqrt{N}$ and
converge to semi-circular behavior, with the remaining $k$ of size $N$ and
converging to hollow Gaussian ensembles). We generalize their work to consider
non-Hermitian ensembles with complex eigenvalues; instead of a blip new
behavior is seen, ranging from multiple satellites to annular rings. These
results are based on moment method techniques adapted to the complex plane as
well as analysis of singular values, and we further isolate the singular value
joint density formula for the Complex Symmetric Gaussian Ensemble.
