  In this paper we develop a theory for oscillatory integrals with complex
phases. When $f:{\mathbb C}^n \to {\mathbb C}$, we evaluate this phase function
on the basic character ${\rm e}(z) := e^{2\pi i x} e^{2\pi i y}$ of ${\mathbb
C} \simeq {\mathbb R}^2$ (here $z = x+iy \in {\mathbb C}$ or $z = (x,y) \in
{\mathbb R}^2$) and consider oscillatory integrals of the form $$ I \ = \
\int_{{\mathbb C}^n} {\rm e}(f({\underline{z}})) \, \phi({\underline{z}}) \,
d{\underline{z}} $$ where $\phi \in C^{\infty}_c({\mathbb C}^n)$.
  Unfortunately basic scale-invariant bounds for the oscillatory integrals $I$
do not hold in the generality that they do in the real setting. Our main effort
is to develop a perspective and arguments to locate scale-invariant bounds in
(necessarily) less generality than we are accustomed to in the real setting.
