  In this paper, over imaginary quadratic fields, we consider the family of
$L$-functions $L (s, f)$ for an orthonormal basis of spherical Hecke--Maass
forms $f$ with Archimedean parameter $t_f$. We establish asymptotic formulae
for the twisted first and second moments of the central values $L\big(\frac 1
2, f\big)$, which can be applied to prove that at least $33 \%$ of $L\big(\frac
1 2, f\big)$ with $t_f \leqslant T$ are non-vanishing as $T \rightarrow
\infty$. Our main tools are the spherical Kuznetsov trace formula and the
Vorono\"i summation formula over imaginary quadratic fields.
