There's an anachronism when Alexander says they must move on and "turn the page". Books with pages weren't invented for another 200 years - about the time of Julius Caesar. In Alexander's time they rolled up paper in long scrolls.
They show horses equipped with modern-day tack - adjustable headgear with polished brass and chromed rein tips and buckles. More likely in 350 BC they had knotted rope-type harnesses.
This is no clear historical evidence that Alexander and Stateira, Darius' wife, had a sexual relationship. In the series, historian Lloyd Llewellyn-Jones cites Plutarch as saying that Stateira died in childbirth and suggests that the child could not have been Darius', given how long she has been separated from him. Yet Plutarch had also written that Alexander showed no interest in Stateira. Plutarch also said that Darius heard the news about his wife's death from an escaped eunuch, who'd been captured along with the women. The eunuch had defended Alexander, suggesting that "[Darius] should even admire Alexander for having shown greater self-restraint in dealing with Persian women than valor against Persian men."
The series conflates two characters: Sisygambis, Darius' mother, and Stateira, Darius' wife. Sisygambis is never mentioned in the series, even though she was the one who mistakes Hephaestion for Alexander and bows to her. This is depicted in many paintings. Historians believe that Alexander and Sisygambis developed a friendly relationship. Allegedly, when Darius' army came to rescue them, Sisygambis refused to leave. When she heard of Alexander's death, she shut herself in a room and refused food till she died five days later. It is unclear whether this reaction is out of grief or concern about their well-being.