  In four-dimensional symplectic maps complex instability of periodic orbits is
possible, which cannot occur in the two-dimensional case. We investigate the
transition from stable to complex unstable dynamics of a fixed point under
parameter variation. The change in the geometry of regular structures is
visualized using 3D phase-space slices and in frequency space using the example
of two coupled standard maps. The chaotic dynamics is studied using escape time
plots and by computations of the 2D invariant manifolds associated with the
complex unstable fixed point. Based on a normal-form description, we
investigate the underlying transport mechanism by visualizing the escape paths
and the long-time confinement in the surrounding of the complex unstable fixed
point. We find that the escape is governed by the transport along the unstable
manifold across invariant planes of the normal-form.
