From the British Museum:
"Literary sources tell us, though perhaps not reliably, that Alexander (reigned 336-323 BC) chose only a few artists to produce his image, and famous names such as the sculptor Lysippos and the painter Apelles were associated with his portraiture. Though none of the famous images have been recovered, many sculptures in different materials, as well as portraits on gemstones and coins, survive. These were mostly produced long after Alexander's death and while the portraits follow similar general characteristics, they also vary in style.
Alexander was always shown clean-shaven, which was an innovation: all previous portraits of Greek statesmen or rulers had beards. This royal fashion lasted for almost five hundred years and almost all of the Hellenistic kings and Roman emperors until Hadrian were portrayed beardless. Alexander was the first king to wear the all-important royal diadem, a band of cloth tied around the hair that was to become the symbol of Hellenistic kingship.
Earlier portraits of Alexander, in heroic style, look more mature than the portraits made after his death, such as this example. These show a more youthful, though perhaps more god-like character. He has longer hair, a more dynamic tilt of the head and an upward gaze, resembling his description in literary sources.
This head was acquired in Alexandria, the city founded by Alexander in 331 BC, and the location of his tomb. Alexandria was also the capital of the longest surviving Hellenistic dynasty, the Ptolemies. From the time of the reign of Ptolemy I Soter ('Saviour') (305-282 BC), Alexander was worshipped as a god and the forefather of the dynasty."