WI: Philip Hamilton survives his duel

Sort of on a Chernow run right now, and I wonder- what if Alexander Hamilton's eldest son survived (whether or not his father survives as well). By Chernow's account he was a fairly promising young man and given his father's family connections and fame (especially if he dies- death tends to quiet controversies) he could potentially emerge as a major political figure in the Early US if he so chooses, perhaps even one day running for the Presidency.
 
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