We have investigated experimentally and theoretically electronic Raman excitations in GaAs-AlxGa1-xAs quantum wires in magnetic fields. In both the experiments and the calculations we find strong anticrossings of the observed confined magnetoplasmon modes in the vicinity of 2ωc, where ωc is the cyclotron-resonance frequency. The observed splittings and anticrossings arise from internal Coulomb interaction in the one-dimensional electron systems and is found to depend sensitively not only on the wave vector but also on the shape of the confining potential and the mode index of the confined plasmon mode.