  This paper reports the results of an ongoing in-depth analysis of the
classical trajectories of the class of non-Hermitian $PT$-symmetric
Hamiltonians $H=p^2+ x^2(ix)^\varepsilon$ ($\varepsilon\geq0$). A variety of
phenomena, heretofore overlooked, have been discovered such as the existence of
infinitely many separatrix trajectories, sequences of critical initial values
associated with limiting classical orbits, regions of broken $PT$-symmetric
classical trajectories, and a remarkable topological transition at
$\varepsilon=2$. This investigation is a work in progress and it is not
complete; many features of complex trajectories are still under study.
