|an absurd or hopeless pursuit, as of something unattainable|
wild goose chase
A futile search or pursuit, as in I think she sent us on a wild goose chase looking for their beach house. This idiom originally referred to a form of 16th-century horseracing requiring riders to follow a leader in a particular formation (presumably resembling a flock of geese in flight). Its figurative use dates from about 1600.