We consider a model of non-Markovian quantum Brownian motion that consists of a harmonic oscillator bilinearly coupled to a thermal bath, both via its position and via its momentum operators. We derive the master equation for such a model, and we solve the equations of motion for a generic Gaussian system state. We then investigate the resulting evolution of the first and second moments for both an Ohmic and a super-Ohmic spectral density. In particular, we show that, irrespective of the specific form of the spectral density, the coupling with the momentum enhances the dissipation experienced by the system, accelerating its relaxation to the equilibrium as well as modifying the asymptotic state of the dynamics. Eventually, we characterize explicitly the non-Markovianity of the evolution using a general criterion which relies on the positivity of the master equation coefficients.
