Graduate Real vs complex spherical harmonics for hexagonal symmetry

[Junaidjami]

TL;DR

Orbital analysis of magnetic anisotropy energy using second order perturbation theory for hexagonal symmetry

Are real spherical harmonics better than complex spherical harmonics for hexagonal symmetry, which are
directly associated to a finite Lz?

 

[vanhees71]

An eigenvector of $L_z$ in position representation is a standard complex spherical harmonic, i.e.,
$$\text{Y}_{lm}=P_{lm}[\cos(\vartheta)] \exp(\mathrm{i} m \varphi).$$
Note that in spherical coordinates the position representation of $\hat{L}_z$ reads
$$\hat{L}_z=-\mathrm{i} \hbar \partial_{\varphi}.$$

 

[Junaidjami]

An eigenvector of $L_z$ in position representation is a standard complex spherical harmonic, i.e.,
$$\text{Y}_{lm}=P_{lm}[\cos(\vartheta)] \exp(\mathrm{i} m \varphi).$$
Note that in spherical coordinates the position representation of $\hat{L}_z$ reads
$$\hat{L}_z=-\mathrm{i} \hbar \partial_{\varphi}.$$

Is there any relation between the crystal symmetry and real/complex spherical harmonics? And is there a way to judge the superiority of one over the other?