We study the dynamics of a two-level impurity embedded in a two-dimensional Bose-Hubbard (BH) model at zero temperature from an open quantum system perspective. Results for the decoherence across the whole phase diagram are presented, with a focus on the critical region close to the transition between superfluid and Mott insulator. In particular we show how the decoherence and the deviation from a Markovian behaviour are sensitive to whether the transition is crossed at commensurate or incommensurate densities. The role of the spectrum of the BH environment and its non-Gaussian statistics, beyond the standard independent boson model, is highlighted. Our analysis resorts on a recently developed method (2020 Phys. Rev. Res. 2 033276) - closely related to slave boson approaches - that enables us to capture the correlations across the whole phase diagram. This semi-analytical method provides us with a deep insight into the physics of the spin decoherence in the superfluid and Mott phases as well as close to the phase transitions.
