  In a previous paper, the author examined the possible torsions of an elliptic
curve over the quadratic fields $\mathbb Q(i)$ and $\mathbb Q(\sqrt{-3})$.
Although all the possible torsions were found if the elliptic curve has
rational coefficients, we were unable to eliminate some possibilities for the
torsion if the elliptic curve has coefficients that are not rational. In this
note, by finding all the points of two hyperelliptic curves over $\mathbb Q(i)$
and $\mathbb Q(\sqrt{-3})$, we solve this problem completely and thus obtain a
classification of all possible torsions of elliptic curves over $\mathbb Q(i)$
and $\mathbb Q(\sqrt{-3})$.
