Why did Alexander lead his army across the gedrosian Dessert.

Alexander led his army across the Gedrosian Desert because that seemed the easiest passage toward India.

Explanation:

After Alexander managed to conquer Persia with his army, he continued with his military conquests, toward east and northeast. The geography of the area east and northeast of Persia was not the best when it comes to military conquests.

India for example, a desired land by Alexander, had excellent natural protection by very high and harsh mountain ranges, such as the Hindu Kush, Pamir, and the Himalayas, as well as the Gedrosian Desert. The Gedrosian Desert, despite being very harsh and inhospitable place, seemed to be far better option for crossing than going 5,000 to 8,000 meters up the enormous mountains, so that's what the Macedonian army did. As they crossed the Gedrosian Desert they got to northwestern India and started their military campaign.