We investigate martingales appropriate for use in complex Banach spaces in connection with the complex uniform convexity popularized
by Davis, Garling and Tomczak. This brings us into contact with diverse concepts, such as: pseudo-convex sets, plurisubharmonic
functions, conformal martingales, the Radon-Nikodym property, and the analytic Randon-Nikodym property.
Supported in part by N.S.F. grant DMS84-01986.