  We introduce the notion of rationality for hyperholomorphic functions
(functions in the kernel of the Cauchy-Fueter operator). Following the case of
one complex variable, we give three equivalent definitions: the first in terms
of Cauchy-Kovalevskaya quotients of polynomials, the second in terms of
realizations and the third in terms of backward-shift invariance. Also
introduced and studied are the counterparts of the Arveson space and Blaschke
factors.
