The historian Arnold Toynbee, a classicist who was the son-in-law of
another famous classicist, published "What If
Alexander the
Great Had
Lived On?" as part of his book "Some Problems of Greek History."
Recognising the enormous spread of Hellenic culture--carried all the
way to Britain by Roman legions, carried all the way to Japan by Indian
and Chinese Buddhisat missionaries--Toynbee sought to create a world
where the Hellenic civilization's political spread equals its cultural
spread.
As in your TL, Alexander recovers, but while he's recovering from his
illness, a triumvirate of his advisors govern on his behalf (birth of
constitutional monarchy).
Alexander circumnavigates the Arabian peninsula, and has the
Phoenicians establish trading cities on its Red Sea and Persian Gulf
shores, but doesn't really bother with the interior.
In the Mediterranean, Alexander allies with Rome and conquers Carthage.
In the east, both India and China are conquered. Mahayana Buddhism
(the Buddhism practiced by China, Korea, and Japan) becomes the
universal religion. After the conquests of India and China, the World
State expands outward peacefully.
Africa is circumnavigated both from the east (by the Phoenicians) and
the west (by the Carthaginians). Hannibal discovers Atlantis (the New
World), and Hero's invention of the steam engine enables rail travel to
become possible at the time that Christ would have been a little boy.
The world ruler, c. 1968, when the essay was published, was Alexander
LVI. Politically, the world state is made up of federations of city
states.