We have studied the optical signatures of the Mg acceptor in GaN, using samples that are doped with Mg during MOCVD growth. In order to reduce the defect density in the material and thus achieve narrow linewidths in optical spectra we have used thick HVPE grown GaN layers as templates in the MOCVD growth. The photoluminescence (PL) spectra show two acceptor-related bound exciton peaks at 3.466 eV and 3.455 eV respectively. In the lower photon energy range the 3.27 eV emission with its LO-phonon replicas is dominant, riding on a broad background emission peaking at about 3.1 eV. These results, together with previous data in the literature, indicate that there are two acceptors in Mg-doped GaN, one dominating the optical spectra (the 3.466 eV and the 3.27 eV emissions) and another related to the 3.455 eV and the 3.1 eV emissions. We suggest that the latter is related to the Mg acceptor, while the former is a H-related complex, not necessarily involving Mg.