  Quantum chaotic systems with one-dimensional spectra follow spectral
correlations of orthogonal (OE), unitary (UE), or symplectic ensembles (SE) of
random matrices depending on their invariance under time reversal and rotation.
In this letter, we study the non-Hermitian and non-unitary ensembles based on
the symmetry of matrix elements, viz. ensemble of complex symmetric, complex
asymmetric (Ginibre), and self-dual matrices of complex quaternions. The
eigenvalues for these ensembles lie in the two-dimensional plane. We show that
the fluctuation statistics of these ensembles are universal and quantum chaotic
systems belonging to OE, UE, and SE in the presence of a dissipative
environment show similar spectral fluctuations. The short-range correlations
are studied using spacing ratio and spacing distribution. For long-range
correlations, unfolding at a non-local scale is crucial. We describe a generic
method to unfold the two-dimensional spectra with non-uniform density and
evaluate correlations using number variance. We find that both short-range and
long-range correlations are universal. We verify our results with the quantum
kicked top in a dissipative environment that can be tuned to exhibit symmetries
of OE, UE, and SE in its conservative limit.
