The stability of the complex-rotation method in B-spline basis for a simple atomic resonant scattering problem in free field is investigated. The numerical calculation shows that this method has a feature that the solution will not change in a wide range of rotation angle θ. Our determined scattering resonant energies and widths exactly coincide with the popularly accepted values. A new resonance is identified numerically although it is very broad.The norm of the complex eigenvalue, [E], is proposed to investigate and to evaluate the stability of the obtained complex eigenvalues.