  The correspondence principle asserts that quantum mechanics resembles
classical mechanics in the high-quantum-number limit. In the past few years
many papers have been published on the extension of both quantum mechanics and
classical mechanics into the complex domain. However, the question of whether
complex quantum mechanics resembles complex classical mechanics at high energy
has not yet been studied. This paper introduces the concept of a local quantum
probability density $\rho(z)$ in the complex plane. It is shown that there
exist infinitely many complex contours $C$ of infinite length on which $\rho(z)
dz$ is real and positive. Furthermore, the probability integral $\int_C\rho(z)
dz$ is finite. Demonstrating the existence of such contours is the essential
element in establishing the correspondence between complex quantum and
classical mechanics. The mathematics needed to analyze these contours is subtle
and involves the use of asymptotics beyond all orders.
