I'd take issue saying crossing the Alps was the most celebrated achievements. It's the fact that he crossed the Alps, losing half(?) his force in the process, then proceed to win three major victories against the Romans, culminating in the Battle of Cannae.

Most military historians consider the Battle of Cannae to be his greatest achievement- in which Hannibal's smaller folder encircled and destroyed the larger Roman one. Several members of the Roman elite, including Senators and Consuls were killed in the battle.