  The complex Busemann-Petty problem asks whether origin symmetric convex
bodies in C^n with smaller central hyperplane sections necessarily have smaller
volume. The answer is affirmative if n\leq 3 and negative if n\geq 4. In this
article we show that the answer remains the same if the volume is replaced by
an "almost" arbitrary measure. This result is the complex analogue of
Zvavitch's generalization to arbitrary measures of the original real
Busemann-Petty problem.
