Abstract: It is well known that coherent states  provide approximate solutions to the time dependent Schrodinger equation in the semiclassical limit if the centre and the shape of the state evolve according to the Hamiltonian dynamics of the corresponding classical system.   We study the case that the Hamilton operator is no longer self-adjoint and find that coherent states still provide approximate solutions, but that the underlying classical system which describes the motion of the center and the shape of the state is no longer Hamiltonian. Instead the dynamics is governed by the combination of a Hamiltonian vectorfield and a gradient vectorfield which are coupled by a time dependent metric. We will concentrate in particular on the quadratic case and show how the above dynamics is related to a projection of the complexified Hamiltonian dynamics and a special complex structure on phase space. This is joint work with Eva-Maria Graefe.