The purpose of this talk is to present some recent advances in the phase space analysis of quadratic differential operators modeled by the Weyl quantization of a complex-valued quadratic symbol. The first part will focus on metaplectic and Schroedinger propagators, where Gabor wave packet decompositions lead to a structured representation which cap- tures insightful phenomena (such as sparsity, dispersion, spreading, confinement). In the second part, we will discuss how these methods provide a unified framework also for non-selfadjoint quadratic models, where heat-type dissipative effects play a crucial role, emphasizing how a phase space perspective effectively connects all the relevant features of the problem.