wild-goose chase(n = noun.act) - the fruitless pursuit of something unattainable;
is a kind of pursuance, pursuit, quest

N  caprice, fancy, humor, whim, whimsy, whimsey, whimwham, crotchet, capriccio, quirk, freak, maggot, fad, vagary, prank, fit, flimflam, escapade, boutade, wild-goose chase, capriciousness, kink, capricious, erratic, eccentric, fitful, hysterical, full of whims, maggoty, inconsistent, fanciful, fantastic, whimsical, crotchety, kinky, particular, humorsome, freakish, skittish, wanton, wayward, contrary, captious, arbitrary, unconformable, penny wise and pound foolish, fickle, frivolous, sleeveless, giddy, volatile, by fits and starts, without rhyme or reason, nil fuit unquain sic inipar sibi, the deuce is in him.

chase, chase after, chase away, give chase, paper chase, salmon p. chase, salmon portland chase, wild goose chase