Most Battle of Midway what if's focus on the Japanese doing something different and smarter with their carrier based forces or the Americans not getting so lucky, but what if at Midway after the loss of the four Japanese carriers, Yamamoto had gone with his first instinct and sent his battleships all out to engage the American forces?
Now, I know that he chose not to because the Americans had complete command of the air.
But weren't the American carrier based squadrons severely depleted after the battle? The torpedo squadrons were effectively wiped out and IIRC at least two of the dive bomber squadrons were decimated (though they had succeeded in taking out three of the Japanese carriers).
Wouldn't it have been possible for the Japanese battleship forces to have closed with the withdrawing American carrier based forces and engaged in a significant surface engagement with them?