What is the kinematics of a particle with complex mass?

I think the question has no meaningful answer, at least in our universe. If you look at $$E^2 - p^2 = m^2$$ then if $m$ is complex with non-zero real and imaginary components, then $m^2$ is also complex with non-zero real and imaginary components and therefore either $E$ or $p$ (or both) must also be complex with non-zero real and imaginary components. I don't think there is any meaningful description of the kinematics of a particle with complex energy or momentum.