  We propose a measure, which we call the dissipative spectral form factor
(DSFF), to characterize the spectral statistics of non-Hermitian (and
non-Unitary) matrices. We show that DSFF successfully diagnoses dissipative
quantum chaos, and reveals correlations between real and imaginary parts of the
complex eigenvalues up to arbitrary energy (and time) scale. Specifically, we
provide the exact solution of DSFF for the complex Ginibre ensemble (GinUE) and
for a Poissonian random spectrum (Poisson) as minimal models of dissipative
quantum chaotic and integrable systems respectively. For dissipative quantum
chaotic systems, we show that DSFF exhibits an exact rotational symmetry in its
complex time argument $\tau$. Analogous to the spectral form factor (SFF)
behaviour for Gaussian unitary ensemble, DSFF for GinUE shows a
"dip-ramp-plateau" behavior in $|\tau|$: DSFF initially decreases, increases at
intermediate time scales, and saturates after a generalized Heisenberg time
which scales as the inverse mean level spacing. Remarkably, for large matrix
size, the "ramp" of DSFF for GinUE increases quadratically in $|\tau|$, in
contrast to the linear ramp in SFF for Hermitian ensembles. For dissipative
quantum integrable systems, we show that DSFF takes a constant value except for
a region in complex time whose size and behavior depends on the eigenvalue
density. Numerically, we verify the above claims and additionally compute DSFF
for real and quaternion real Ginibre ensembles. As a physical example, we
consider the quantum kicked top model with dissipation, and show that it falls
under the universality class of GinUE and Poisson as the `kick' is switched on
or off. Lastly, we study spectral statistics of ensembles of random classical
stochastic matrices or Markov chains, and show that these models fall under the
class of Ginibre ensemble.
