  We introduce a complex-extended continuum level density and apply it to
one-dimensional scattering problems involving tunneling through finite-range
potentials. We show that the real part of the density is proportional to a real
"time shift" of the transmitted particle, while the imaginary part reflects the
imaginary time of an instanton-like tunneling trajectory. We confirm these
assumptions for several potentials using the complex scaling method. In
particular, we show that stationary points of the potentials give rise to
specific singularities of both real and imaginary densities which represent
close analogues of excited-state quantum phase transitions in bound systems.
