  Consider the multiplication operator $M_{B}$ in $L^2(\T)$, where the symbol
$B$ is a finite Blaschke product. In this article, we characterize the
commutant of $M_{B}$ in $L^2(\T)$, noting the fact that $L^2(\T)$ is not an
RKHS. As an application of this characterization result, we explicitly
determine the class of conjugations commuting with $M_{z^2}$ or making
$M_{z^2}$ complex symmetric by introducing a new class of conjugations in
$L^2(\T)$. Moreover, we analyze their properties while keeping the whole Hardy
space, model space, and Beurling-type subspaces invariant. Furthermore, we
extended our study concerning conjugations in the case of finite Blaschke.
