The term 'Hamilton Faceoff' is a reference to the famous DUEL between Alexander Hamilton and Aaron Burr in 1804. The crossword solution for this term is 'DUEL', which is a four letter word meaning a formal fight or contest between two people, usually involving weapons. This term is particularly relevant to the Hamilton-Burr DUEL, which was a deadly confrontation between the two men that resulted in Hamilton's death.