Alexander wanted to unite all of Persia under his rule and mix the non-Greek cultures with the Greek. Therefore, his vision, at least as far as it could be implemented, was characterized mainly by an interest in disseminating and preserving Greek culture, as opposed to replacing or destroying cultures. He married two Persian princesses and encouraged his generals to do the same. Greek architects and sculptors were brought to construct his new cities. Two of the greatest ancient libraries built at Alexandria and Pergamum were the home to hundreds of thousands of texts and attracted all sorts of scholars and artists. The Greeks also adopted Eastern beliefs and customs. The fusion of cultures gave rise to a new dialect known as Koine, Greek. Egyptian gods were worshipped under the Ptolemaic dynasty. However, after Alexander died his empire quickly divided into four parts around 300 BC due to competing power struggles amongst his generals. Those kingdoms would all be occupied by Rome.
References:
https://en.wikipedia.org/wiki/Hellenistic_period
Glencoe World History, Chapter 4 Ancient Greece