With weight-shifting operators, $d\neq 2$ looks increasingly like $d=2$ in CFT

When working on conformal field theory, your life is very different depending on whether the dimension is two or not. In $d=2$ you have that infinite-dimensional symmetry algebra called the Virasoro algebra, and in some important cases such as minimal models you can classify your CFTs, and solve them analytically. In $d\neq 2$, your symmetry algebra is finite-dimensional, and you mostly have to do with numerical results. This not only makes you code a lot, but also incites you to make technical assumptions that are physically restrictve, such as unitarity.

Degenerate fields in $d=2$ CFT

What makes $d=2$ CFT solvable in many cases is the existence of degenerate primary fields.