  We study the Loewner evolution whose driving function is $W_t = B_t^1 + i
B_t^2$, where $(B^1,B^2)$ is a pair of Brownian motions with a given covariance
matrix. This model can be thought of as a generalization of Schramm-Loewner
evolution (SLE) with complex parameter values. We show that our Loewner
evolutions behave very differently from ordinary SLE. For example, if neither
$B^1$ nor $B^2$ is identically equal to zero, then the set of points
disconnected from $\infty$ by the Loewner hull has non-empty interior at each
time. We also show that our model exhibits three phases analogous to the phases
of SLE: a phase where the hulls have zero Lebesgue measure, a phase where
points are swallowed but not hit by the hulls, and a phase where the hulls are
space-filling. The phase boundaries are expressed in terms of the signs of
explicit integrals. These boundaries have a simple closed form when the
correlation of the two Brownian motions is zero.
