Growth of torsion of elliptic curves with full 2-torsion over quadratic cyclotomic fields

Let K=Q(−3) or Q(−1) and let Cn 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]≈C2⊕C2 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.