Taking the Gaussian vortex beam as an example, the variation of electric and magnetic polarization singularities of focused Gaussian vortex beams is studied. It is shown that in the focal region there exist two- and three-dimensional (2D and 3D) electric and magnetic polarization singularities of focused Gaussian vortex beams, which do not coincide in general. By suitably varying the distance from the focal plane and the truncation parameter, the motion, creation, and annihilation of circular polarization singularities will take place. The critical valuse of the parameters at which a pair of circular polarization singularities of 2D and 3D electric and magnetic fields will annihilate, are not the same. In the 2D electric field the V-point may appear at the focal plane.