We investigate from first principles the energetic and vibrational properties of various candidate structures for the 3125 cm-1 local vibrational mode in GaN, known to be related to hydrogen passivated magnesium atoms. The orientation of the electric dipole of this mode has recently been measured with respect to the wurtzite c-axis, giving a result seemingly inconsistent with current atomic models for this defect. We study the possibility that complexes of magnesium, native impurities and hydrogen could give rise to the experimental observations. Furthermore, we consider a possible candidate giving rise to a 0.88-eV line in a variety of electron-irradiated GaN samples. We find evidence that a deep donor level including substitutional oxygen must result from a complex impurity.