In this paper, we study complex Wishart processes or the so-called Laguerre processes (Xt)t≥0. We are interested in the behaviour of the eigenvalue process; we derive some useful stochastic differential equations and compute both the infinitesimal generator and the semi-group. We also give absolute-continuity relations between different indices. Finally, we compute the density function of the so-called generalized Hartman–Watson law as well as the law of T0:=inf {t, det (Xt)=0} when the size of the matrix is 2.