The complex study of the polarization-controlled supercontinuum generation in a dual-core square lattice photonic crystal fiber made of multicomponent glass was accomplished. The fiber was excited by 100-fs pulses at a 1250-nm wavelength in the anomalous dispersion region and the registered spectra exhibited soliton fission and Raman-induced self-frequency shift processes. The study also involved a detailed analysis of the infrared-to-visible light conversion exhibiting a dispersive wave origin. The special dual-core properties were investigated by separate spectral analysis of each of the two cores and the near-field profile registration. The emphasis was on the visible part of the spectrum where the input energy and polarization direction dependences were studied. The increase in the input energy allowed for the tuning of the wavelength of the visible spectral features, a further rotation of the polarization direction had an effect on the spectral dependence of the light distribution between the cores. The dual-core fiber exhibited a significant coupling performance and the spectral dependence of the visible light distribution is in good agreement with the simulated coupling length spectral characteristics. Single-core excitation in the linear regime revealed the possibility of coupling 50% of the energy to the other core and the same polarization-controlled redirection possibilities as that at the nonlinear experiments. Dual-core excitation of the fiber enhances the light redirection effect with the application potential for the polarization-controlled directional coupler accompanied by nonlinear frequency conversion.