The domain may be obtained as follows
Reduce[(4 - x^4) >= 0, x] -Sqrt[2] <= x <= Sqrt[2]
Another approach, which uses the definition of domain, is
In[3]:= Reduce[res == (4 - x^4)^(1/4), {x, res}, Reals] Out[3]= -Sqrt[2] <= x <= Sqrt[2] && res == (4 - x^4)^(1/4)
