  In the context of two illustrative examples from supersymmetric quantum
mechanics we show that the semi-classical analysis of the path integral
requires complexification of the configuration space and action, and the
inclusion of complex saddle points, even when the parameters in the action are
real. We find new exact complex saddles, and show that without their
contribution the semi-classical expansion is in conflict with basic properties
such as positive-semidefiniteness of the spectrum, and constraints of
supersymmetry. Generic saddles are not only complex, but also possibly
multi-valued, and even singular. This is in contrast to instanton solutions,
which are real, smooth, and single-valued. The multi-valuedness of the action
can be interpreted as a hidden topological angle, quantized in units of $\pi$
in supersymmetric theories. The general ideas also apply to non-supersymmetric
theories.
