  Bifurcations of periodic orbits as an external parameter is varied are a
characteristic feature of generic Hamiltonian systems. Meyer's classification
of normal forms provides a powerful tool to understand the structure of phase
space dynamics in their neighborhood. We provide a pedestrian presentation of
this classical theory and extend it by including systematically the periodic
orbits lying in the complex plane on each side of the bifurcation. This allows
for a more coherent and unified treatment of contributions of periodic orbits
in semiclassical expansions. The contribution of complex fixed points is find
to be exponentially small only for a particular type of bifurcation (the
extremal one). In all other cases complex orbits give rise to corrections in
powers of $\hbar$ and, unlike the former one, their contribution is hidden in
the ``shadow'' of a real periodic orbit.
