I don't think "purifiers" cool the air, at least no more than a fan does. And a fan doesn't actually "cool" the air, just the person. Now, there may be combination purifiers/air conditioners, for all I know, but I bet they'd have to be installed someway.
There are roll-around portable air conditioners that are free-standing and draw the air from the inside of the room and then vent to the outside through a large tube stuck in a sort of window thingy (sorry, can't get more technical than "thingy".....) I've seen them and they're pretty bulky and pretty loud. Not to mention, I don't think they're cheap. But because they're drawing the air from INSIDE a house, it has to be cleaner than what is coming from outside. Is that what you mean by a "portable" AC?