On one's high horse is an idiom that means someone is being arrogant, haughty, or pretentious. There are several synonyms we can use instead of this phrase. One can say that someone is acting superior, being boastful, or showing off their superiority complex. Other alternatives include being snobbish, acting conceited, or displaying an inflated ego. One can also use terms like pompous, haughty, or overbearing to describe such behavior. In short, there are many phrases that convey the same meaning as on one's high horse, and one can choose the most suitable alternative from among these synonyms depending on the situation.