  The Euclidean path integral quite often involves an action that is not
completely real {\it i.e.} a complex action. This occurs when the Minkowski
action contains $t$-odd CP-violating terms. Analytic continuation to Euclidean
time yields an imaginary term in the Euclidean action. In the presence of
imaginary terms in the Euclidean action, the usual method of perturbative
quantization can fail. Here the action is expanded about its critical points,
the quadratic part serving to define the Gaussian free theory and the higher
order terms defining the perturbative interactions. For a complex action, the
critical points are generically obtained at complex field configurations. Hence
the contour of path integration does not pass through the critical points and
the perturbative paradigm cannot be directly implemented. The contour of path
integration has to be deformed to pass through the complex critical point using
a generalized method of steepest descent, in order to do so. Typically, what is
done is that only the real part of the Euclidean action is considered, and its
critical points are used to define the perturbation theory. In this article we
present a simple 0+1-dimensional example, of $N$ scalar fields interacting with
a U(1) gauge field, in the presence of a Chern-Simons term, where
alternatively, the path integral can be done exactly, the procedure of
deformation of the contour of path integration can be done explicitly and the
standard method of only taking into account the real part of the action can be
followed. We show explicitly that the standard method does not give a correct
perturbative expansion.
