  The papers \cite{O1,O2} are the first works to apply the theory of fiber
bundles in the study of the quaternionic slice regular functions. The main goal
of the present work is to extend the results given in \cite{O1}, where the
quaternionic right linear space of quaternionic slice regular functions was
presented as the base space of a fiber bundle. When the quaternionic right
linear space of quaternionic slice regular functions is associated to certain
domains then this paper shows that the elements of total space, given in
\cite{O1}, are defined from a pair of harmonic functions and a pair of
orthogonal vectors. Simplifying the computations presented in \cite{O1}, where
each element of the total space is formed by two pair of conjugate harmonic
functions and a pair of orthogonal unit vectors.
  This work also gives some interpretations of the behavior of the zero sets of
some quaternionic slice regular polynomials
  in terms of the theory of fiber bundles.
