The Extension->Automatic
does not work. I hope this will improve in later versions. You can give the extension manually:
Factor[a^2 - 3 b^2, Extension -> Sqrt[3]]
or have it computed by Mathematica itself:
pol = a^2 - 3 b^2;
Factor[pol,
Extension -> Cases[Solve[pol == 0],
_?NumericQ, All]]