  In this paper we follow up the study of 'complex complex landscapes,' rugged
landscapes of many complex variables. Unlike real landscapes, the
classification of saddles by index is trivial. Instead, the spectrum of
fluctuations at stationary points determines their topological stability under
analytic continuation of the theory. Topological changes, which occur at
so-called Stokes points, proliferate among saddles with marginal (flat)
directions and are suppressed otherwise. This gives a direct interpretation of
the gap or 'threshold' energy -- which in the real case separates saddles from
minima -- as the level where the spectrum of the hessian matrix of stationary
points develops a gap. This leads to different consequences for the analytic
continuation of real landscapes with different structures: the global minima of
'one step replica-symmetry broken' landscapes lie beyond a threshold, their
hessians are gapped, and are locally protected from Stokes points, whereas
those of 'many step replica-symmetry broken' have gapless hessians and Stokes
points immediately proliferate. A new matrix ensemble is found, playing the
role that GOE plays for real landscapes in determining the topological nature
of saddles.
