Preperiodic points for quadratic polynomials over cyclotomic quadratic fields

Given a number field $K$ and a polynomial $f(z) in K[z]$ of degree at least 2, one can construct a finite directed graph $G(f,K)$ whose vertices are the $K$-rational preperiodic points for $f$, with an edge $alpha to beta$ if and only if $f(alpha) = beta$. Restricting to quadratic polynomials, the dynamical uniform boundedness conjecture of Morton and Silverman suggests that for a given number field $K$, there should only be finitely many isomorphism classes of directed graphs that arise in this way. Poonen has given a conjecturally complete classification of all such directed graphs over $mathbb{Q}$, while recent work of the author, Faber, and Krumm has provided a detailed study of this question for all quadratic extensions of $mathbb{Q}$. In this article, we give a conjecturally complete classification like Poonens, but over the cyclotomic quadratic fields $mathbb{Q}(sqrt{-1})$ and $mathbb{Q}(sqrt{-3})$. The main tools we use are dynamical modular curves and results concerning quadratic points on curves.