Short anwser- we don't know

Long anwser:

Garthwaites "The Persians" (as well as other, more Greek focused literature) speculate that Alexander wanted to form a combined Greco-Persian culture. The Persians would prove throughout their history that their culture was exceedingly influential. The Arab and Mongol invasions prove this point well. Both of them adopted facets of the Persian court and civic administration.

Cultural diffusion was present even in ancient times, and was especially prevalent in the areas of Anatolia. Even so their cultures were radically different, and Alexander had a large army- a weak Persia made a good (read rich) target. Garthwaite argues that because Alexander was quick to marry off his generals to Persian princesses, as well as adopt some Persian coronation rituals that Alexander was planning on uniting the two regions. Burning of Persepolis notwithstanding.

However, some historians- like Homa Katouzian in his book "A History of Persia" argue that Alexander had no desire to unite the Macedonian Greek and Persian cultures, and simply wished to quickly establish his legitimacy in the eyes of the native people. The quickest way to do this was to affect local coronation rituals.

Historically, the sources we have say that Alexander did not wish to kill Darius- whether to prevent his martyrdom, to capture and parade him in some sort of Proto-triumphal fashion, or simply out of professional courtesy we don't know.

As an aside, the 10th century epic poem "Shahnahmeh" by Abolquasem Ferwdowsi (a staple in Persian cultural literature) has Iskandr (Alexander) weeping over the corpse of Darius and promising to bring his murderers to justice- Garthwaites historiography seems to line up with this (purely anecdotal, probably fictional) account of things.