  The quantum evolution of the Wigner function for Gaussian wave packets
generated by a non-Hermitian Hamiltonian is investigated. In the semiclassical
limit $\hbar\to 0$ this yields the non-Hermitian analog of the Ehrenfest
theorem for the dynamics of observable expectation values. The lack of
Hermiticity reveals the importance of the complex structure on the classical
phase space: The resulting equations of motion are coupled to an equation of
motion for the phase space metric---a phenomenon having no analog in Hermitian
theories.
