The spin s=(1,0) Bose symmetry of the Dirac equation for the free spinor field, proved recently in our papers, is extended for the Dirac equation interacting with external Coulomb field. Relativistic hydrogen atom is modeling here by such Dirac equation. We are able to present both the fermionic and bosonic symmetries known from our papers about the case of non-interacting spinor field. New symmetry operators were found on the basis of new gamma matrix representations of the Clifford and SO(8) algebras, which were found recently in our papers. Hidden symmetries were found both in the canonical Foldy-Wouthuysen and in the covariant Dirac representations. The symmetry operators, which are simple and graceful in the Foldy-Wouthuysen representation, become non-local in the Dirac model.