  Positive $T$-martingales were developed as a general framework that extends
the positive measure-valued martingales and are meant to model intermittent
turbulence. We extend their scope by allowing the martingale to take complex
values. We focus on martingales constructed on the interval $T=[0,1]$ and
replace random measures by random functions. We specify a large class of such
martingales for which we provide a general sufficient condition for almost sure
uniform convergence to a nontrivial limit. Such a limit yields new examples of
naturally generated multifractal processes that may be of use in multifractal
signals modeling.
