The most appropriate meaning of the idiom "wild goose chase" is futile search.

This idiom means a search that is completely unsuccessful and a waste of time because the person or thing being searched for does not exist or is somewhere else. It is often used to describe a situation where someone has been given incorrect information and is therefore wasting their time looking for something that does not exist.

The other options are not correct.

• Doing something crazy: This is not the same as a wild goose chase.
• Working very hard for something: This is not the same as a wild goose chase.
• To betray someone: This is not the same as a wild goose chase.