  Complex trajectories for Hamiltonians of the form H=p^n+V(x) are studied. For
n=2 time-reversal symmetry prevents trajectories from crossing. However, for
n>2 trajectories may indeed cross, and as a result, the complex trajectories
for such Hamiltonians have a rich and elaborate structure. In past work on
complex classical trajectories it has been observed that turning points act as
attractors; they pull on complex trajectories and make them veer towards the
turning point. In this paper it is shown that the poles of V(x) have the
opposite effect --- they deflect and repel trajectories. Moreover, poles shield
and screen the effect of turning points.
