Prior to his death, Alexander planned on invading Arabia, that is a maratime, based expedition to subjugate the coastal kingdoms of Arabia, while marching all the way inland through the desert would have been logistically unsound. As for the western Mediterranean, he may have wanted to create a new military hegemony with the Greek colonies in Megale Hellas/Magna Graecia against the naval dominance of Carthage. Rome in the late Fourth Century BCE was really just one of the main city-state leagues in Italy and was still competing against the Samnite League, and was in no position, during the 320's BCE, to be challenging either Carthage or the Macedonian Empire.