  We introduce a complex generalization of Wigner time delay $\tau$ for
sub-unitary scattering systems. Theoretical expressions for complex time delay
as a function of excitation energy, uniform and non-uniform loss, and coupling,
are given. We find very good agreement between theory and experimental data
taken on microwave graphs containing an electronically variable lumped-loss
element. We find that time delay and the determinant of the scattering matrix
share a common feature in that the resonant behavior in $\text{Re}[\tau]$ and
$\text{Im}[\tau]$ serves as a reliable indicator of the condition for Coherent
Perfect Absorption (CPA). This work opens a new window on time delay in lossy
systems and provides a means to identify the poles and zeros of the scattering
matrix from experimental data. The results also enable a new approach to
achieving CPA at an arbitrary energy/frequency in complex scattering systems.
