  Let $f_c(z) = z^2+c$ for $c \in \mathbb{C}$. We show there exists a uniform
bound on the number of points in $\mathbb{P}^1(\mathbb{C})$ that can be
preperiodic for both $f_{c_1}$ and $f_{c_2}$ with $c_1\not= c_2$ in
$\mathbb{C}$. The proof combines arithmetic ingredients with complex-analytic;
we estimate an adelic energy pairing when the parameters lie in
$\bar{\mathbb{Q}}$, building on the quantitative arithmetic equidistribution
theorem of Favre and Rivera-Letelier, and we use distortion theorems in complex
analysis to control the size of the intersection of distinct Julia sets. The
proof is effective, and we provide explicit constants for each of the results.
