  In this article, we study some potential theoretical and topological aspects
of the generalized Mandelbrot set introduced by Baker and DeMarco. For $\alpha$
real, we study the set of all totally real algebraic parameters $c$ such that
$\alpha$ is preperiodic under the iteration of the one-parameter family $f_c(x)
= x^2 + c$. We show that when $|\alpha| < 2$ and rational then the set of
totally real algebraic parameters $c$ with this property is finite, whereas if
$|\alpha| \geq 2$ and rational then this set is countably infinite. As an
unexpected consequence of this study, we also show that when $|\alpha| \geq 2$
then parameters $c$ such that $\alpha$ is $f_c$-periodic are necessarily real.
As a special case, we classify all totally real algebraic integers $c$ such
that $\alpha = \pm1$ is preperiodic.
