special relativity - What is the kinematics of a particle with complex mass?

• particles with real-mass have time-like kinematics ($ds^2 > 0$).


• particles with zero-mass have light-like kinematics ($ds^2 = 0$).


• particles with imaginary-mass have space-like kinematics ($ds^2 < 0$) (tachyons).

So the question is pretty simple:

What would be the kinematics of a particle with mass that has both non-zero real and imaginary parts?

Answer

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.