  Let $K = \mathbb{Q}(\sqrt{-3})$ or $\mathbb{Q}(\sqrt{-1})$ and let $C_n$
denote the cyclic group of order $n$. We study how the torsion part of an
elliptic curve over $K$ grows in a quadratic extension of $K$. In the case
$E(K)[2] \approx C_1$ we investigate how a given torsion structure can grow in
a quadratic extension and the maximum number of extensions in which it grows.
We also study the torsion structures which occur as the quadratic twist of a
given torsion structure. In order to achieve this we examine $N$-isogenies
defined over $K$ for $N=15,20,21,24,27,30,35$.
