  Non-Hermitian quantum systems exhibit fascinating characteristics such as
non-Hermitian topological phenomena and skin effect, yet their studies are
limited by the intrinsic difficulties associated with their eigenvalue
problems, especially in larger systems and higher dimensions. In Hermitian
systems, the semiclassical theory has played an active role in analyzing
spectrum, eigenstate, phase, transport properties, etc. Here, we establish a
complex semiclassical theory applicable to non-Hermitian quantum systems by an
analytical continuation of the physical variables such as momentum, position,
time, and energy in the equations of motion and quantization condition to the
complex domain. Further, we propose a closed-orbit scheme and physical meaning
under such complex variables. We demonstrate that such a framework
straightforwardly yields complex energy spectra and quantum states, topological
phases and transitions, and even the skin effect in non-Hermitian quantum
systems, presenting an unprecedented perspective toward nontrivial
non-Hermitian physics, even with larger systems and higher dimensions.
