The story of Alexander's death in Babylon is a bit mysterious, because our sources mention a "Royal diary" that is not very well-known. However, the information taken from it seems sound, even though there are some very strange elements in it.
Alexander died on 11 June 323 BCE, in the late afternoon; this can be deduced from the Astronomical Diaries, a Babylonian source. Several scholars have argued for 13 June and 10 June, but the first of these dates is based on an inaccurate Greek source that uses a confused Egyptian calendar, and the second is based on inaccurate reading of the Astronomical diary.
The following text is taken from the Anabasis by the Greek author Arrian of Nicomedia (7.24.4-27.2), translated by Aubrey de Sélincourt.
The death of Alexander
[7.24.4] A few days later Alexander was sitting at dinner with his friends and drinking far into the night. He had previously celebrated the customary sacrificial rites in thanks for his success, adding certain others in obedience to his seers' advice, and had also, we are told, distributed wine and sacrificial victims among the various units and sections of the army. According to some accounts, when he wished to leave his friends at their drinking and retire to his bedroom, he happened to meet Medius, who at that time was the companion most closely in his confidence, and Medius asked him to come and continue drinking at his own table, adding that the party would be a merry one.
[7.25.1] The Royal diaries confirm the fact that he drank with Medius after his first carouse. Then (they continue) he left the table, bathed, and went to sleep, after which he supped with Medius and again set to drinking, continuing till late at night. Then, once more, he took a bath, ate a little, and went straight to sleep, with the fever already on him.
[7.25.2] Next day he was carried out on his bed to perform his daily religious duties as usual, and after the ceremony lay in the men's quarters till dark. He continued to issue orders to his officers, instructing those who were to march by landnote[Alexander planned an expedition to southern Arabia, the country where incense was produced.] to be ready to start in three days and those who were going with himself by sea to sail one day later.
[7.25.3] From he was carried on his bed to the river,note[The Euphrates.] and crossed in a boat to the park on the further side, where he took another bath and rested. Next day he bathed again and offered sacrifice as usual, after which he went to lie down in his room, where he chatted to Medius and gave orders for his officers to report to him early next morning.
[7.25.4] Then he took a little food, returned to his room, and lay all night in a fever.
The following morning he bathed and offered sacrifice, and then issued to Nearchusnote[Alexander's admiral.]and the other officers detailed instructions about the voyage, now due to start in two days' time.
Next day he bathed again, went through regular religious duties, and was afterwards in constant fever.
[7.25.5] None the less he sent for his staff as usual and gave them further instructions on their preparations for sailing. In the evening, after another bath, his condition was grave, and the following morning he was moved to the building near the swimming-pool. He offered sacrifice, and, in spite of his increasing weakness, sent for his senior officers and repeated his orders for the expedition.
[7.25.6] The day after that he just managed to have himself carried to his place of prayer, and after the ceremony still continued, in spite of his weakness, to issue instructions to his staff.
Another day passed. Now very seriously ill, he still refused to neglect his religious duties; he gave orders, however, that his senior officers should wait in the court, and the battalion and company commanders outside his door. Then, his condition already desperate, he was moved from the park back to the palace. He recognized his officers when they entered his room but could no longer speak to them. From that moment until the end he uttered no word. That night and the following day, and for the next twenty-four hours, he remained in a high fever.
[7.26.1] These details are all to be found in the Diaries. It is further recorded in these documents that the soldiers were passionately eager to see him; some hoped for a sight of him while he was still alive; others wished to see his body, for a report had gone round that he was already dead, and they suspected, I fancy, that his death was being concealed by his guards. But nothing could keep them from a sight of him, and the motive in almost every heart was grief and a sort of helpless bewilderment at the thought of losing their king. Lying speechless as the men filed by, he yet struggled to raise his head, and in his eyes there was a look of recognition for each individual as he passed.
[7.26.2] The Diaries say that Peitho, Attalus, Demophon, and Peucestas, together with Cleomenes, Menidas, and Seleucus, spent the night in the temple of Serapisnote[The cult of Serapis was "invented" in Egypt, more than a generation after the death of Alexander. This proves that the Royal diary is not a contemporary document, but a later fabrication. On the other hand, it should be noted that the compiler was not tempted to add "famous last words".] and asked the god if it would be better for Alexander to be carried into the temple himself, in order to pray there and perhaps recover; but the god forbade it, and declared it would be better for him if he stayed where he was.
[7.26.3] The god's command was made public, and soon afterwards Alexander died - this, after all being the "better" thing.
The accounts of both Ptolemy and Aristobulusnote[Arrian's main sources, discussed here.] end at this point. Other writers have added that the high officers most closely in his confidence asked him to name his successor, and that Alexander's reply was "the best man".note[A play on the name Craterus, "the strongest man".] There is also a story that he went on to say that he knew very well there would be funeral "games" in good earnest after he was dead.note[Another word play: the word for game also means struggle and war.]
[7.27.1] I am aware that much else has been written about Alexander's death: for instance, that Antipater sent him some medicine which had been tampered with and that he took it, with fatal results. Aristotle is supposed to have made up this drug, because he was already afraid of Alexander on account of Callisthenes' death, and Antipater's son Cassander is said to have brought it. Some accounts declare that he brought it in a mule's hoof,
[7.27.2] and that it was given Alexander by Cassander's younger brother Iollas, who was his cup-bearer and had been hurt by him in some way shortly before his death; others state that Medius, who was Iollas' lover, had a hand in it, and support that view by the fact that it was Medius who invited Alexander to the drinking- party - he felt a sharp pain after draining the cup, and left the party in consequence of it.