The chordal Loewner equation provides a correspondence between real-valued functions, called driving functions, and certain growing 2-dimensional sets, called hulls.  In this talk, we will consider the generalization to complex-valued driving functions, which was first studied by Huy Tran.  We will discuss some key differences between the hulls in the complex-valued setting and those in the real-valued setting, including the question of the phase transition from simple-curve hulls.  This is joint work with Jeffrey Utley.