  Hypercomplex signal processing (HSP) provides state-of-the-art tools to
handle multidimensional signals by harnessing intrinsic correlation of the
signal dimensions through Clifford algebra. Recently, the hypercomplex
representation of the phase retrieval (PR) problem, wherein a complex-valued
signal is estimated through its intensity-only projections, has attracted
significant interest. The hypercomplex PR (HPR) arises in many optical imaging
and computational sensing applications that usually comprise quaternion and
octonion-valued signals. Analogous to the traditional PR, measurements in HPR
may involve complex, hypercomplex, Fourier, and other sensing matrices. This
set of problems opens opportunities for developing novel HSP tools and
algorithms. This article provides a synopsis of the emerging areas and
applications of HPR with a focus on optical imaging.
