  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
$E(K)[2] \approx C_2 \oplus C_2$ we determine how a given torsion structure can
grow in a quadratic extension and the maximum number of quadratic extensions in
which it grows. We also classify the torsion structures which occur as the
quadratic twist of a given torsion structure.
