  By analogy with the Lobachevsky space H_{3}, generalized parabolic
coordinates (t_{1},t_{2},\phi) are introduced in Riemannian space model of
positive constant curvature S_{3}. In this case parabolic coordinates turn out
to be complex valued and obey additional restrictions involving the complex
conjugation. In that complex coordinate system, the quantum-mechanical Coulomb
problem is stu- died: separation of variables is carried out and the wave
solutions in terms of hypergeometric functions are obtained. At separating the
variables, two parameters k_{1} and k_{2} are introduced, and an operator B
with the eigen values (k_{1}+k_{2}) is found, which is related to third
component of the known Runge-Lenz vector in space S_{3} as follows: i B = A
_{3} + i \vec{L}^{2}, whereas in the Lobachevsky space as B =A_{3} +
\vec{L}^{2}. General aspects of the possibility to employ complex coordinate
systems in the real space model S_{3} are discussed.
