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We deal with a complex-valued Ornstein-Uhlenbeck (OU) process with parameter$\lambda\in\mathbb{R}$starting from a point different from 0 and the way thatit winds around the origin.The starting point of this paper is the skew productrepresentation for an OU process which is associated to the skew productrepresentationof its driving planar Brownian motion under a new deterministictime scale.We present the stochastic differential equations (SDEs)for theradial and for the winding process. Moreover, we obtain the large time(analogue of Spitzer's Theorem for Brownian motion in the complex plane) andthe small time asymptotics for the winding and for the radialprocess, and weexplore the exit time from a cone for a 2-dimensional OU process.Some LimitTheorems concerning the angle of the cone (when our process winds in a cone)and the parameter $\lambda$ are also presented.Furthermore, we discuss thedecomposition of the winding process of a complex-valued OU process in "small"and "big" windings,where, for the "big" windings, we use some results alreadyobtained by Bertoin and Werner in \cite{BeW94},and we show that only the"small" windings contribute in the large time limit.Finally, we study thewindings of a complex-valued OU process driven by a Stable processand we obtainsimilar results for its (well-defined) winding and radial process.

On the windings of complex-valued Ornstein-Uhlenbeck processes driven by a Brownian motion and by a Stable process