Chasing something that doesn't exist

"Chasing after the wind" sort of flows with that example sentence you provided

Wild goose chase: (via dictionary.com)

Definition:

a wild or absurd search for something nonexistent or unobtainable

Example:

a wild-goose chase looking for a building long demolished

"chasing rainbows" seems like a good choice.

• trying to achieve something that is not possible or practical TFD

e.g. He wanted to go into show business, but friends told him to quit chasing rainbows.

I'm always chasing rainbows
Watching clouds drifting by
My schemes are just like all my dreams
Ending in the sky... (lyrics by Joseph McCarthy)

or

• "to reach for the unreachable star" as in "The Impossible Dream".

e.g. "The problem is I'm always trying to reach the unreachable star."