Self-duality equations in gauge theory can be complexified in many inequivalent ways, but there are two obvious options: One can extend Hodge duality in either a complex linear fashion, or in a conjugate linear one. In general, the two cases result in two very different equations. The first case was first studied by Haydys, while the second is due to Kapustin and Witten.  In this talk, I will focus on the complexification of the 3-dimensional BPS monopole equations.  We call solutions to these equations Haydys and Kapustin-Witten monopoles, respectively. We find a stark contrast between them: On one hand, we construct an open neighborhood of the BPS moduli space within the Haydys moduli space, and show that this neighborhood is a smooth, hyperkahler manifold of dimension twice that of the Bogomolny moduli space. On the other hand, we prove that a (finite energy) Kapustin-Witten monopole is necessarily a BPS monopole when the structure group is SU(2). In fact, this latter result generalizes to 4-dimensions as well. Joint work with Goncalo Oliveira, UFF, Brazil.