let $e$ be an elliptic curve defined over ${q}$. the torsion subgroup of $e$ over the compositum of all quadratic extensions of ${q}$ was studied by michael laska, martin lorenz, and yasutsugu fujita. laska and lorenz described a list of $31$ possible groups and fujita proved that the list of $20$ different groups is complete. in this talk, we will generalize the results of laska, lorenz and fujita to the elliptic curves defined over a quadratic cyclotomic field i.e. $q(i)$ and $q(\sqrt{-3})$.