  In this paper, we study complex Wishart processes or the so-called Laguerre
processes $(X_t)_{t\geq0}$. 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 $T_0:=\inf\{t,\det(X_t)=0\}$ when the size of the matrix is 2.
