  We construct a counterexample to a well-known extension theorem for slice
regular functions, which motivates us to develop a theory of Riemann
slice-domains by introducing a new topology on quaternions. By some paths
describing axial symmetry in Riemann slice-domains, we rectify the classical
extension formula in the theory of slice regular functions and prove a
representation formula over slice-domains of regularity. This proof involves an
intertwining relation between imaginary units of quaternions and a fixed matrix
corresponding to a complex structure.
