  We deal with complex spatial diffusion equations with time-fractional
derivative and study their stochastic solutions. In particular, we complexify
the integral operator solution to the heat-type equation where the time
derivative is replaced with the convolution-type generalization of the
regularized Caputo derivative. We prove that this operator is solution of a
complex time-fractional heat equation with complex spatial variable. This
approach leads to a wrapped Brownian motion on a circle time-changed by the
inverse of the related subordinator. This time-changed Brownian motion is
analyzed and, in particular, some results on its moments, as well as its
construction as weak limit of continuous-time random walks, are obtained. The
extension of our approach to the higher dimensional case is also provided.
