I don't have 5e, but is the Monk of the Four Winds what you want? Despite the name it is general-element-based, not specifically air-based.
What do you want a monk of the four elements to be able do? I'm betting there's some way to reproduce it in PF.
You could build something along those lines (one power at a time) with a Qinggong Monk. I think that's the closest you can come without resorting to 3pp.
OK: unarmed combat I assume, good saves, elemental powers including area effect damage and utility.
First up the qinggong monk archetype. There's many options but only a few are elemental - scorching ray, hydraulic torrent, gaseous form and a few others. You could stretch it a little further with style feats like Marid Style/Spirit/Coldsnap or Efreeti Style/Stance/Touch, but its magic is limited. Also it's powered by ki points which run out very fast in Pathfinder.
Next there's a kineticist, possibly with the elemental ascetic archetype. That archetype requires melee only despite being more monk-like; the base kineticist taking the kinetic fist form infusion for use when punching and other forms might work better. Either is restricted to one element until 7th level, two elements until 15th level, three until 20th. If a monk of the four elements needs more elements earlier that'll be a problem of course.
An magus with the esoteric archetype punches people and is a 2/3 spellcaster with many elemental spell options. I'd thought it not too impressive, but I've seen arguments lately that it can be made to work.
One last possibility is an oracle. The ascetic mystery covers monk flavour, the elemental imbalance curse covers one element (probably air), careful spell selection can cover more, and if you take the spirit guide archetype you can pick one element each day to be really good at.
Edit: Fuzzy, I'm sure those aren't all the options either. Punching people + magic isn't that uncommon a combination in PF and most types of magic include at least some elemental stuff.