  In the article [PT] a general procedure to study solutions of the equations
$x^4-dy^2=z^p$ was presented for negative values of $d$. The purpose of the
present article is to extend our previous results to positive values of $d$. On
doing so, we give a description of the extension ${\mathbb
Q}(\sqrt{d},\sqrt{\epsilon})/{\mathbb Q}(\sqrt{d})$ (where $\epsilon$ is a
fundamental unit) needed to prove the existence of a Hecke character over
${\mathbb Q}(\sqrt{d})$ with fixed local conditions. We also extend some "large
image" results regarding images of Galois representations coming from ${\mathbb
Q}$-curves (due to Ellenberg in \cite{MR2075481}) from imaginary to real
quadratic fields.
