Partial fraction expansion with complex numbers

I found by trial and error that Extension-> Sqrt[I] does the job.

ExpToTrig[Apart[Factor[1/(1 + x^4), Extension -> Sqrt[I]]]]

$$frac{frac{1}{4}+frac{i}{4}}{sqrt{2} left(-x+frac{1+i}{sqrt{2}}right)}+frac{frac{1}{4}+frac{i}{4}}{sqrt{2} left(x+frac{1+i}{sqrt{2}}right)}-frac{frac{1}{4}-frac{i}{4}}{sqrt{2} left(-x-frac{1-i}{sqrt{2}}right)}-frac{frac{1}{4}-frac{i}{4}}{sqrt{2} left(x-frac{1-i}{sqrt{2}}right)}$$

Here ExpToTrig is not really required but it does the final beautifying.