  The phase retrieval problem in the classical setting is to reconstruct
real/complex functions from the magnitudes of their Fourier/frame measurements.
In this paper, we consider a new phase retrieval paradigm in the
complex/quaternion/vector-valued setting, and we provide several
characterizations to determine complex/quaternion/vector-valued functions $f$
in a linear space $S$ of (in)finite dimensions, up to a trivial ambiguity, from
the magnitudes $\|\phi(f)\|$ of their linear measurements $\phi(f), \phi\in
\Phi$. Our characterization in the scalar setting implies the well-known
equivalence between the complement property for linear measurements $\Phi$ and
the phase retrieval of linear space $S$. In this paper, we also discuss the
affine phase retrieval of vector-valued functions in a linear space and the
reconstruction of vector fields on a graph, up to an orthogonal matrix, from
their absolute magnitudes at vertices and relative magnitudes between
neighboring vertices.
