factor(x^4+1) and partfrac(1/(x^4+1))

[rombva]

I check out the tests from the document provided by link.

x^4+1=((x-(1-i)/sqrt(2))*(x-(1+i)/sqrt(2)))*((x-(-1-i)/sqrt(2))*(x-(-1+i)/sqrt(2)))=
=(x^2-sqrt(2)*x+1)*(x^2+sqrt(2)*x+1)

I don't want to get a factorization result over the complex field. The polynomial x^4+1 may be rewriten as a product of polynomials over the field of reals (for example, HP 50g can factorize it). What's wrong with HP Prime's factorization in the real mode in this case?