We consider the following inequality for a complex number $z$ and a real-valued function $f$ : $\left |z-\frac ar\right |<\frac {f(r)}{|r|} ,(a,r) = 1,$ where $a$ and $r$ are integers in an imaginary quadratic field $\mathbf {Q}(\sqrt {d})$, $d < 0$. We denote by $A_f$ the set of $z$ having infinitely many solutions $a/r$ to the above inequality. We show that either $A_f$ or $A_f^c$ is a set of Lebesgue measure $0$ (in the complex plane). We also give a sufficient condition on $f$ so that $A_f^c$ is a set of Lebesgue measure $0$, which is a complex version of Duffin-Schaeffer's condition.