WordNet sense 1 (a search for an alternative that meets cognitive criteria):

pursuance, quest

WordNet sense 2 (an auxiliary activity):

avocation, hobby, sideline, by-line, spare-time activity

WordNet sense 3 (a diversion that occupies one's time and thoughts (usually pleasantly)):

interest, pastime

WordNet sense 4 (the act of pursuing in an effort to overtake or capture):

chase, following, pursual