How does chemical bonding look like on the quantum scale per se since the electron is a wave, and a particle?
I believe that generally in chemistry the wave characteristics of the electron are far more prevalent/relevant that the particle characteristics. So it's usually easiest just to think of it as a wave (or I think "region of space" is easier).
A chemical bond looks like the electron wave or area spread out between two or more atoms. This shape of the wave has lower energy than any of the orbitals it could occupy on just one atom. Because it has lower energy, that means it needs energy from somewhere for the electrons & atoms to change back to a different configuration. So unless it gets some more energy it cannot change away from the bonded state. That explains why chemical bonds are stable and hold atoms together.