  Dito and Turrubiates recently introduced an interesting model of the
dissipative quantum mechanics of a damped harmonic oscillator in phase space.
Its key ingredient is a non-Hermitian deformation of the Moyal star product
with the damping constant as deformation parameter. We compare the
Dito-Turrubiates scheme with phase-space quantum mechanics (or deformation
quantization) based on other star products, and extend it to incorporate Wigner
functions. The deformed (or damped) star product is related to a complex
Hamiltonian, and so necessitates a modified equation of motion involving
complex conjugation. We find that with this change the Wigner function
satisfies the classical equation of motion. This seems appropriate since
non-dissipative systems with quadratic Hamiltonians share this property.
