Deliberate mistake: When Alexander is giving his pre-battle speech at Gaugamela, some of the shots are 'flipped', which has the odd effect of suddenly making all of the phalanx pikemen left-handed.
Deliberate mistake: Throughout the film, scenes of several Cole Porter productions are shown being performed. These scenes all contain a well-known musical number. Two scenes in particular (after the Alanis Morissette number and immediately after the "So in Love" duet) end with the lead singer/singers bowing to the audience and then waving the other cast members on stage to take a company bow. The problem is that neither of those numbers appear at the end of the shows being "performed." The "So in Love" duet, for example, most definitely appears in the musical "Kiss Me, Kate" well before the Act Two finale. I assume that these mistakes were filmed deliberately to emphasize the importance of the songs in question, but they are still errors.
Deliberate mistake: During the flight Hughes takes with Katherine Hepburn, they talk to each other in normal, conversational voices. A twin-piston-engined aircraft of that vintage (or any vintage for that matter) is LOUD. We should be able to hear roaring engines, and the two should have to shout at each other to be heard.