With Compton scattering the total momentum and the total energy of the photon electron system is conserved, so it has to be ELASTIC Collision/Scattering.
But an argument is that just because the kinetic energy of the incoming photon is different than the outgoing photon, it is INELASTIC Collision/Scattering.
However, I thought we had to look at the system, which is the photon and electron interaction; not just the scattered photon itself.
So, what is the real issue here?