  The dynamics generated by non-Hermitian Hamiltonians are often less intuitive
than those of conventional Hermitian systems. Even for models as simple as a
complexified harmonic oscillator, the dynamics for generic initial states shows
surprising features. Here we analyse the dynamics of the Husimi distribution in
a semiclassical limit, illuminating the foundations of the full quantum
evolution. The classical Husimi evolution is composed of two factors, (i) the
initial Husimi distribution evaluated along phase-space trajectories, and (ii)
the final value of the norm corresponding to each phase-space point. Both
factors conspire to lead to intriguing dynamical behaviours. We demonstrate how
the full quantum dynamics unfolds on top of the classical Husimi dynamics for
two instructive examples.
