  Infinite order differential operators appear in different fields of
mathematics and physics. In the past decade they turned out to play a crucial
role in the theory of superoscillations and provided new insight in the study
of the evolution as initial data for the Schr\"odinger equation. Inspired by
the infinite order differential operators arising in quantum mechanics, in this
paper we investigate the continuity of a class of infinite order differential
operators acting on spaces of entire hyperholomorphic functions. Precisely, we
consider homomorphisms acting on functions in the kernel of the Dirac operator.
For this class of functions, often called monogenic functions, we introduce the
proximate order and prove some fundamental properties. As important application
we are able to characterize infinite order differential operators that act
continuously on spaces of monogenic entire functions.
