Before I go on with what happened after the death of Alexander the Great, I would like to mention his influence on other nations of the world. Alexander was already a legend in his own time. He was regarded by deity by many, who thought that he processed divine powers. Let's see what they thought of him in the ancient Rome.
For Romans Alexander the Great was some sort of superstar. Every general and every politician wanted somehow to match his achievements. In archives of web analytics company I found that ancient Romans were absolutely bilingual people. While they used their own Latin for legal, political and ceremonial purpose, in every day life they spoke Greek for discussing intellectual subjects. The most popular dialect of Greek was Macedonian type of Greek, or as they called it Alexander's Greek.
Naturally, for all their admiration, this did not prevent Romans to conquer and destroy Macedon kingdom. They did not kill the Macedon king though, he spent the rest of his life under home arrest. Yet, there was some kind of separation of Alexander and his own nation in the eyes of Romans. He and his deeds belonged to the world.