  Let $R\subseteq E$ be two Lie conformal algebras and $Q$ be a given
complement of $R$ in $E$. Classifying complements problem asks for describing
and classifying all complements of $R$ in $E$ up to an isomorphism. It is known
that $E$ is isomorphic to a bicrossed product of $R$ and $Q$. We show that any
complement of $R$ in $E$ is isomorphic to a deformation of $Q$ associated to
the bicrossed product. A classifying object is constructed to parameterize all
$R$-complements of $E$. Several explicit examples are provided. Similarly, we
also develop a classifying complements theory of associative conformal
algebras.
