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Hamilton had always been volatile, though Burr used to refer to it as something more like "passionate". And of course, he swept Burr along every time. Hamilton was the one who lured Burr into the more romantic side of their relationship, though it was short-lived once the two became increasingly on more opposing sides in the political world. Burr could never separate the two like Hamilton could, but even though their relationship was short lived Burr could admit (if only to himself) that he never stopped loving Hamilton and his fiery, albeit vexing, personality.
But he can't think of that now, now that he and Hamilton are standing across the field from one another with pistols at their sides. He'd like to blame this on Hamilton too, that he was a victim of the hurricane that is Alexander, but he knew this was his doing. This was his pride. And now this is his life, Theodosia's future, which is at risk. He cannot back down, regardless of how much he wants to have Hamilton in his life rather than Hamilton's blood on his hands. He supposes that his reluctance to take action to stop the battle is karma from his public indifference in policy.
But Hamilton! Hamilton could serve to be a little more indifferent. It was his words, his actions that lead them here. It was Hamilton's words that kept him from achieving the presidency. His friend, his lover, chose Jefferson over him. Now, this is the result.
Though, to be fair Burr does not picture himself walking out of this duel. He was in the army, just as Hamilton was, but he never did learn mastery of a weapon. Honestly, he probably couldn't hit Hamilton given three tries even if the fog wasn't obstructing his view.
The seconds have started the count. Burr raises his pistol, does as he is told. The count reached ten and Burr squeezes his eyes shut as he squeezes the trigger.
Looking back on that moment Burr realizes that what he notices first isn't the lack of pain or even the recoil of the gun. It's Alexander's scream that follows.
He drops his weapon and runs. For all his preparation for the battle Burr never expected to win. He didn't want to think of a world where Hamilton was dead, especially one where it was by Burr's own hand.
By the time he reaches Alexander he knows that he will survive. Alexander was clutching his shoulder, blood pouring out between his fingers and teeth clenched but still standing. Again, Burr is swept away in Hamilton. He almost collides with him from how fast he was running. He immediately grabs Alexander's face in his hands and crashes their lips together. Not practical or in any way subtle – especially in full view of William and Hamilton's crew – but that is what Alexander makes him. Impulsive.
Burr is brought back to realty as Hamilton releases a pained breath. He releases Hamilton, calling over the doctor who had kept his distance after Burr's… interaction with Hamilton. Hamilton smiles at him, even while clutching the evidence of Burr's murderous intent. Hamilton was odd in that way, never letting "political matters" interfere with their relationship.
"Do you yield?" Hamilton manages as the doctor makes his way over. It takes Burr a second to realize what he was asking, the original purpose of the duel fleeing his mind as soon as he heard Alexander's scream.
"Yes, Alexander. I yield."
He stays near Hamilton as the doctor binds his shoulder, and he follows when they retreat to the nearby hospital for further treatment. Burr stays through the cleaning and patching of the wound. And when Hamilton is released to go home and rest, Burr follows.