  In this paper we begin the study of Schur analysis and de Branges-Rovnyak
spaces in the framework of Fueter hyperholomorphic functions. The difference
with other approaches is that we consider the class of functions spanned by
Appell-like polynomials. This approach is very efficient from various points of
view, for example in operator theory, and allows to make connections with the
recently developed theory of slice polyanalytic functions.
  We tackle a number of problems: we describe a Hardy space, Schur multipliers
and related results. We also discuss Blaschke functions, Herglotz multipliers
and their associated kernels and Hilbert spaces. Finally, we consider the
counterpart of the half-space case, and the corresponding Hardy space, Schur
multipliers and Carath\'eodory multipliers.
