While the complement of a real arrangement A is topologically simple (the only interesting invariant is the number of connected components, or regions of the arrangement), the complexified complement M(A) is much richer. There, the cohomology ring is isomorphic to the Orlik—Solomon algebra of the underlying matroid and the Poincare polynomial is closely related to the combinatorial characteristic polynomial of the arrangement. In this talk, we look at two models for the complexified complement of an arrangement intersected with a convex set (a setting which naturally arises in many places, including the theory of reflection arrangements and their generalizations). We will define at the complexified complement of such a pair, then look at the analogues of both the much-celebrated Salvetti complex of an arrangement and a less well-known non-Hausdorff model for the complexified complement due to Proudfoot. This work is joint with Dan Dugger and Nick Proudfoot.