  Generalizing the work of Kobayashi and the second author for elliptic curves
with supersingular reduction at the prime $p$, B\"uy\"ukboduk and Lei
constructed multi-signed Selmer groups over the cyclotomic
$\mathbb{Z}_p$-extension of a number field $F$ for more general non-ordinary
motives. In particular, their construction applies to abelian varieties over
$F$ with good supersingular reduction at all the primes of $F$ above $p$. In
this article, we scrutinize the case in which $F$ is imaginary quadratic, and
prove a control theorem (that generalizes Kim's control theorem for elliptic
curves) of multi-signed Selmer groups of non-ordinary motives over the maximal
abelian pro-$p$ extension of $F$ that is unramified outside $p$, which is the
$\mathbb{Z}_p^2$-extension of $F$. We apply it to derive a sufficient condition
when these multi-signed Selmer groups are cotorsion over the corresponding
two-variable Iwasawa algebra. Furthermore, we compare the Iwasawa
$\mu$-invariants of multi-signed Selmer groups over the
$\mathbb{Z}_p^2$-extension for two such representations which are congruent
modulo $p$.
