  In these proceedings I review the status of an ongoing project that aims at
solving the quark propagator Dyson-Schwinger equation in the complex domain.
The novel aspect of the approach is that the non-analyticities arising
throughout the iteration of the equation are to be taken into account in an
appropriate way through dynamic contour deformation. Because of the complexity
of the approach, these studies are undertaken in a heavily truncated scenario
that serves as a toy model for the development of the numerical techniques that
are required to treat the system in a mathematically sound way.
