  We study a characterization of slice Carleson measures and of Carleson
measures for the both the Hardy spaces $H^p(\mathbb B)$ and the Bergman spaces
$\mathcal A^p(\mathbb B)$ of the quaternionic unit ball $\mathbb B$. In the
case of Bergman spaces, the characterization is done in terms of the axially
symmetric completion of a pseudohyperbolic disc in a complex plane. We also
show that a characterization in terms of pseudohyperbolic balls is not
possible.
