Fig.: searching for answers that can't be found, referring to a futile and hopeless undertaking

The police realized that the false witness had misled them, and they were just chasing shadows

La polizia si rese conto che il falso testimone li aveva depistati e quindi stava solo dando la caccia a delle ombre

The project was a complete failure: everybody involved did something wrong and to try and find out who was wrong in the first place would be going on a wild goose chase

Il progetto è fallito miseramente; tutti, chi più chi meno, hanno commesso errori e cercare di scoprire chi per primo ha sbagliato sarebbe come dare la caccia alle ombre