Are there any archetypes/prestige classes/etc. for a paladin that would make him or her more monk-like? Or for a monk that would make him or her more paladin-like? And are there any prestige classes that essentially require a paladin/monk combo to qualify?
Paizo only please!
Thanks!
Enlightened paladin and champion of Irori prestige class
I think the enlightened champion is a better choice.
Note that the correct name for the "enlightened paladin" is "Iroran Paladin" if you are referencing either the Archives of Nethys or Inner Sea Combat (the original source of this archetype).
Both of the options above are flavored for worshipers of Irori.
Nothing for non-Iroran monks?
sacred fist WP is a divine casting archetype that is like a monk.
Iroran paladin/enlightened paladin is one of the more playable examples of this combo.
Mostly because you can throw out the half assed attempt at doing unarmed build. You keep all of your martial weapons, and the only thing that ties you strongly to unarmed strikes is the bonded weapon (which could be nice to keep for a reach build).
It adds CHA along with dex for AC, and you can do it with light armor. So it actually does a light armor character in a very interesting manner- even modest investment in those stats (14) can let you hit the max dex of your armor. You can then invest mostly in a str build with no worries- you get enough AC that you are basically in heavy armor. Only with more touch AC and less flat footed AC.
Your smite is also not tied to alignment, but it is changed into a more mild +1 to +7. But hey- you get ki that you can use to ignore ANYTHING's AC. Perfect for fighting golems and other traditionally sticky enemies.
You keep other tanky stuff- cha to saves, lay on hands, spells, etc. So overall, you can make a fine monk-y paladin with this. You can be a kung-fu swordsman with this going on about how 'the heavens will not forgive your misdeeds!'