  In an anisotropic medium, the refractive index depends on the direction of
propagation. Zero index in a fixed direction implies a stretching of the wave
to uniformity along that axis, reducing the effective number of dimensions by
one. Here we investigate two dimensional gyrotropic media where the refractive
index is zero in a complex valued direction, finding that the wave becomes an
analytic function of a single complex variable z. For simply connected media
this analyticity implies unidirectional propagation of electromagnetic waves,
similar to the edge states that occur in photonic 'topological insulators'. For
a medium containing holes the propagation is no longer unidirectional. We
illustrate the sensitivity of the field to the topology of the space using an
exactly solvable example. To conclude we provide a generalization of
transformation optics where a complex coordinate transformations can be used to
relate ordinary anisotropic media to the recently highlighted gyrotropic ones
supporting one-way edge states.
