I'm studying atomic orbitals and the shape is usually represented with real form spherical harmonics, taken as an appropriate linear combination of the complex ones. If, however, the physical quantity is the probability density, which is the square of the absolute value and thus always real, why don't we just use this quantity and instead represent orbitals as $p_x$, $p_y$, etc?
I'm sorry if this was already answered, but I didn't find anything regarding this specific thing.
As pointed out in the previous answer, both real and complex spherical harmonics form an appropriate basis which an be used to decompose any state of interest. So mathematically speaking they should be equivalent.
Why use one versus the other? I personally prefer to always use the complex versions because atomic states are typically represented in this basis. This is because these states are irreducible representations of the rotation symmetry group so, for example, if you put on a magnetic field to split the energy levels of a system with a magnetic moment you will find that the states represented by complex spherical harmonics are still eigenfunctions of the system and have fixed energies whereas the real spherical harmonics are now no longer eigenstates.
Why then do we often see visualizations of the real spherical harmonics? In chemistry it is understood that the phase of the wavefunction is important for bonding. The real spherical harmonics can either be positive or negative, that is all there is to the phase. This can easily be visualized with two colors on the orbitals and while a little weird can at least be visualized and understood.
To visualize the phase of the complex orbitals you have to use a phase gradient colormap (like in the figure on wikipedia). To understand this colormap you really need a full understanding of what it means for a complex number to have a phase and understand how that might lead to interference etc. In short.. for a high school student trying to understand chemical bonds it would be very difficult to understand what is going on with the complex orbitals. They don't even know what complex numbers are yet!
So in short, I think the complex orbitals are better once you understand complex numbers and can understand phase map visualizations. I think the reason the real orbitals are often shown is because not everyone who cares about atomic orbitals necessarily knows about complex numbers.
Complex spherical harmonics (SH) are not more physical than the real spherical harmonics, and vice versa. They are just used to represent the angular part of whatever common basis functions are chosen. The complex SH has the advantage of being an eigenfunction of rotation operator around the z-axis and that's why they are very convenient to use for single atomic system, i.e. hydrogen-like atoms, since a single atom needs only one center/origin. On the other hand, the real SH are commonly used when you are dealing with molecules for at least two reasons. First, your system has lost spherical symmetry, so it's unproductive to represent the angular part of all of the basis functions relative to a common origin, hence people have been using atom-centered basis. But this means, you no longer need the rotational symmetry of complex SH, which leads to the second reason - the numerical cost of complex numbers. Since complex numbers require twice as many memory space as real ones, while at the same time your wavefunction can be real (assuming there is no external magnetic field, if there is, the wavefunction is complex), it becomes clear that real SH is the one to go here.
