If Alexander did not die in 323 bc, and live another 10-20 years, would he have nextconquered North Africa and Italy, thus preventing the rise of the Carthage and more interestingly, the Roman empires?

I don't think so, for many reasons.

1) His generals were not willy to fight other wars.
2) His macedonian armies as well. Besides, Alexander won against weak enemies: Persians were ridiculous. Persia was a giant with feet of clay. Even with an incredible numerical superiority Persians were not able to defeat one only city, Athens.
3) If Alexander wanted to conquer Cartagho, he needed first to obtain a naval superiority: and this aim was almost impossible to achieve.
4) Pyrrus, the king of Epirus that led a "macedonian-equipped-trained" army in Italy, lost his campaign against Romans at the beginnins of the III century. Roman legions were more flexible than macedonian phalanx.

Alexander the Great (356-323 BC) was compelled to stop the conquest of the Indian sub-continent and to start on the return journey to his Macedonian capital Pella in Northern Greece by his mutinous and campaign-weary soldiers. He died on the way home. At that time he was just 33 years old and still had the desire to seek more glory, as evidenced by his desire to continue his campaigning.

The rich Greek cities of Magna Graecia in Southern Italy would have been a next logical target. This would have definitely brought him on a collision course with Rome, which at that time was not the power it became less than 100 years later.

If not hard historical evidence, at least there are good reasons to think he would have turned his attentions West to Italy. Just a few years after Alexander's death, King Pyrrhus (319-272 BC) of Epirus - the North-West region of Greece adjacent to Macedonia - campaigned in Italy against Rome, and in fact won every battle he fought against the Romans. He withdrew from Italy as his "logistics" could not keep up with the resources and resilience of the Roman armies. With access to their local resources, the Romans were able to recover from their battle losses. Hence a hollow "Pyrrhic" victory still bears Pyrrhus' name. Note that Alexander's mother Olympias was an Epirote Greek princess. King Pyrrhus may have been related to Alexander.

Alexander would have been able to overcome Pyrrhus' "logistics" and lack of re-inforcements problems by using the manpower and wealth of the immense empire he had conquered. Also at his time Rome was not as powerful as it was later in Pyrrhus' time.

Had Alexander expanded to and consolidated the Greek world and Hellenism in Italy, the history of Europe and the World would have evolved differently. For better or for worse is a whole different argument worthy of many Ph.D. theses

There's nothing really left for me to say after the two previous posts.

As interesting an idea as that would have been, it would not have happened.
If the plan of Alexander, his generals and the Macedonian army had been to take on the war hawks of the Roman Republic, they almost certainly would have struck them first, before beginning a long, wearying campaign in Asia Minor. But such a move would not have been logical nor smart.

The only aim the Macedonian army had when they began their march was to strike back and conquer Asia Minor in return for the humiliating invasion of Greece by the Persian king Xerxes a century before. They wanted to leave a peaceful Mediterranean behind them, with no warring city states nor a hostile Rome to cause trouble in their absence.