  Generalised Hagedorn wave packets appear as exact solutions of Schr\"odinger
equations with quadratic, possibly complex, potential, and are given by a
polynomial times a Gaussian. We show that the Wigner transform of generalised
Hagedorn wave packets is a wave packet of the same type in phase space. The
proofs build on a parametrisation via Lagrangian frames and a detailed analysis
of the polynomial prefactors, including a novel Laguerre connection. Our
findings directly imply the recently found tensor product structure of the
Wigner transform of Hagedorn wave packets.
