I have a field $K$ and an idempotent element $\alpha\not\in K$ (i.e., $\alpha^2=\alpha$), and I would like to form the ring $K[\alpha]$. Can this structure exist (1) in general and (2) with $K=\mathbb{C}$?
It's one thing to mod out $\alpha^2+1$ but modding out (the reducible) $\alpha^2-\alpha$ seems very unusual. (At the risk of repeating myself, $0\ne\alpha\ne1$.)
You're thinking of constructing $K[X]/(X^2-X)$ This is just $K\times K$ by the Chinese Remainder Theorem. That's as general as it gets, since any other structure you're thinking factors through this. This is not a field, of course, since it has nontrivial idempotents, but fields, in general domains, only have the trivial ones: $0,1$.