We show an exact equivalence of the free energy of the q-state Potts antiferromagnet on a lattice for the full temperature interval and the free energy of the q-state Potts model on the dual lattice for a semi-infinite interval of complex temperatures (CTs). This implies the existence of two quite different types of CT singularities: the generic kind, which does not obey universality or various scaling relations, and a special kind which does obey such properties and encodes information of direct physical relevance. We apply this observation to characterize CT properties of the Potts model on several lattices, to rule out two existing conjectures, and to determine the critical value of q above which the Potts antiferromagnet on the diced lattice has no phase transition.