chase after vs quest

chase after is not a noun while quest is a noun.

Word	Noun	Adjective	Verb	Adverb
chase after	No	No	Yes	No
quest	Yes	No	Yes	No

As verbs, quest is a hyponym of chase after; that is, quest is a word with a more specific, narrower meaning than chase after:

Other hyponyms of chase after include tree, hound, hunt, trace, run down.

chase after (noun)	quest (noun)
	the act of searching for something
	a search for an alternative that meets cognitive criteria

chase after (verb)	quest (verb)
go after with the intent to catch	seek alms, as for religious purposes
pursue someone sexually or romantically	express the need or desire for
	bark with prolonged noises, of dogs
	search the trail of (game)
	make a search (for)