Counterfactuals abound in this. What if Stamford Bridge had failed?

What if Harold had won?

Likely, the long-term outcome would be the same. Harald wouldn't have been in a great position to fight of William, either. He might have simply ceded southern England without even a fight.

And given William's subsequent Harrying of the North, I don't think that Harald would be able to hold on to it very long. He was intent on taking over the whole thing, and he had a much stronger force coming over the English Channel than over the North Sea.

Of course we can never really tell. I could spin a million other possible outcomes. But in this case, William really did have an overwhelming force. Godwinson had a home-field advantage, and might have won if he hadn't exhausted his force just getting there. The Vikings weren't going to fare better.

I find one of the more interesting counterfactuals to be "What if William gets better winds and lands in late August?"