A vexed question refers to a matter that is highly controversial or problematic. When searching for synonyms for vexed question, one might consider using phrases like a thorny issue, a hot potato, or a difficult problem. Other options include a contentious matter, a polarizing topic, or a disputable concern. Some might refer to it as a divisive issue, a bothersome dilemma, or a contentious debate. Regardless of the phrasing, these synonyms all convey the complex and challenging nature of a topic that presents considerable difficulty and controversy.