He is trying to give too much physical interpretation to the various components of the wavefunction. The phase of the wavefunction (which is usually where the complex part is at play) basically only affects what energy levels the molecule has. In the molecular orbital picture, we can treat the wavefunction as made up of many molecular orbitals interfering with each other. The magnetic properties of the molecule meanwhile depends on the whole wave function, not just the complex part as the author claimed