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As you clearly outlined, ##\gamma_l^m## has complex values on a sphere. In the case of ##p_z## there is no problem because ##\gamma## is...

Those look like the real and imaginary parts of the spherical harmonic.

Wavefunctions are usually written down in a coordinate system where the z-axis is singled out, i.e. the azimuthal quantum number ## m ##...

The Spherical harmonics are in the form ##\gamma_l^{m}(\theta, \phi) \propto P_l^{m}(cos \theta) e^{im\phi}##...

I think you agree that the p-orbitals are spherical harmonics ##\gamma_l^{m}(\theta, \phi) \propto P_l^{m}(cos \theta) e^{im\phi}## with...