Although we are not sure what language Hannibal actually used, it stands to reason that he was proficient with the most common language spoken in the Carthage and in 3rd century BCE this would have been Punic. This language, now extinct, was derived from the Phoenician and is considered a variant thereof, with Phoenician and Punic belonging to the Caananite group of the Northwest Semitic language family, along with Amorite, Hebrew and Moabite with some scholars also including Ugaritic and Edomite. Nevertheless, all these languages are extinct today with the sole exception of Hebrew that survived as the religious language of the Jews.

It seems that from the languages used today, Hebrew would be the closest approximation to the Phoenician and Punic with a possibility that a proficient Hebrew user would be more or less able to understand at least some of the written Phoenician or Punic after making necessary adjustment for the difference in the script used (both Hebrew and Phoenician used abjad, i.e. form of script that notes only consonants as separate symbols). For example, the name of the rendered in Latin as 'Kartago' is accepted to be rendition of Punic Qart Hadasht that bears a close similarity to modern Hebrew 'קרת חדשת', both meaning 'New City'. Similarly, Phoencian inscription found of the sarcophagus of the king Tabnit reads in Latin transcription: "'nk tbnt khn 'shtrt mlk sdnm bn 'shmn'zr" (I, Tabnit, priest of Ashtarte, king of Sidon, son of Ashmenezar), while the modern Hebrew would be quite similar ('ni tbnt kwhn 'shtrt mlch sidwn bn 'shmnwzr). Existing Phoenician inscription show similarity to Hebrew in both lexis and grammar. By the way, the name of the Hannibal's family (Barkids) is derived form the name Barka(s) cognate with the Semitic root brq, also found in the modern Arabic and Hebrew name 'Barak'.

So, to sum it up, Hannibal most likely spoke Punic that, from all the popular languages spoken in 21st century, would be closest to modern Hebrew, with at least some similarity to other Semitic languages, like Arabic.

Great answer, thanks for sharing!