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In this article, we study the set of parameters $c \in \mathbb{C}$ for whichtwo given complex numbers $a$ and $b$ are simultaneously preperiodic for thequadratic polynomial $f_{c}(z) = z^{2} +c$. Combining complex-analytic andarithmetic arguments, Baker and DeMarco showed that this set of parameters isinfinite if and only if $a^{2} = b^{2}$. Recently, Buff answered a question oftheirs, proving that the set of parameters $c \in \mathbb{C}$ for which both$0$ and $1$ are preperiodic for $f_{c}$ is equal to $\lbrace -2, -1, 0\rbrace$. Following his approach, we complete the description of these setswhen $a$ and $b$ are two given integers with $\lvert a \rvert \neq \lvert b\rvert$.

Simultaneously preperiodic integers for quadratic polynomials