We study the statistical properties of the complex generalization of Wigner time delay τW for subunitary wave-chaotic scattering systems. We first demonstrate theoretically that the mean value of the Re[τW] distribution function for a system with uniform absorption strength η is equal to the fraction of scattering matrix poles with imaginary parts exceeding η. The theory is tested experimentally with an ensemble of microwave graphs with either one or two scattering channels and showing broken time-reversal invariance and variable uniform attenuation. The experimental results are in excellent agreement with the developed theory. The tails of the distributions of both real and imaginary time delay are measured and are also found to agree with theory. The results are applicable to any practical realization of a wave-chaotic scattering system in the short-wavelength limit, including quantum wires and dots, acoustic and electromagnetic resonators, and quantum graphs.