  We derive a semiclassical approximation for the evolution generated by the
Lindblad equation as a generalization of complex WKB theory. Linear coupling to
the environment is assumed, but the Hamiltonian can be a general function of
positions and momenta. The theory is carried out in the chord representation
and describes the evolved quantum characteristic function, which gives direct
access to the Wigner function and the position representation of the density
operator by Fourier transforms. The propagation is shown to be of Liouville
type in a complex double phase space, the imaginary part of the action being
responsible for decoherence. The theory is exact in the quadratic case, just as
the real WKB theory that we previously developed for the Markovian evolution of
extended states, but it also describes the decoherent and dissipative evolution
of localized states, such as the interference terms of a Schr\"odinger cat
state. The present rederivation of the real WKB approximation leads to its
interpretation as a first order classical perturbation of the complex theory
and to a discussion of its validity. The example of a simple cubic Hamiltonian
illustrates the various levels of approximation derived from the complex WKB
theory.
