- 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.
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