  We determine the set of integral points on elliptic curves of rank $2$
defined over imaginary quadratic fields using quadratic Chabauty. This builds
on the work of Balakrishnan, Bianchi, Besser and M\"uller, and gives the first
example of quadratic Chabauty used to determine the integral points on a curve
which is not a base change from $\mathbb{Q}$.
