  Let $H$ be a separable complex Hilbert space. A conjugate-linear map $C:H\to
H$ is a conjugation if it is an involutive isometry. In this paper, we consider
the following interpolation problems: Let $\{x_i\}_{i\in I}$ and $\{y_i\}_{i\in
I}$ be two orthogonal sets of vectors in $H$, and let $N$ and $\{N_k\}_{k\in
K}$ be normal operators such that the $N_k$'s mutually commute. Then, under
which conditions does there exist a conjugation $C$ on $H$ such that
  (a) $Cx_i=y_i$ and $CN_kC=N_k^*$ for all $i\in I$ and $k\in K$; or
  (b) $Cx_i=y_i$, for every $i\in I$, and $CNC=-N^*$.
  Using the spectral projections of normal operators, we provide complete
answers to problems (a) and (b). As an application, we characterize
hyperinvariant subspaces of normal operators using conjugations. Additionally,
we give necessary and sufficient conditions for the existence of solutions to
certain equations in $L^{\infty}(\mu)$.
