How is this cartridge removed?

Just a guess is that a set screw in the new cartridge at the top? That could be the answer.

You should be able to unscrew the indices at the top, by hand, and then there should be a cross head screw. Unscrew, pull and the cartridge/handle should detach relatively easily. Out of interest where did you buy the replacement cartridge?

Do you see the top cap (in the picture, it looks like the ridged edge of a U.S. quarter dollar coin)? It probably says "Hot" or "Cold" or something like that, or maybe nothing. That may simply unscrew out of the handle, under which there will likely be a stem screw that threads into the spline.

Looks expensive. What brand is it?

quarter turn? probably ceramic disk with screen. Ceramic disks will last forever of treated right. They are terrible for taking shock and tend to break when slammed quickly. But im just speculating on what it is.