Answer
$$y=\sqrt{\sqrt{x}-1},\:y=-\sqrt{\sqrt{x}-1},\:y=\sqrt{-\sqrt{x}-1},\:y=-\sqrt{-\sqrt{x}-1}$$
Work Step by Step
Switching the elements of the domain and the range (in other words, the dependent and independent variables), we find that the inverse is:
$$x=\left(1+y^2\right)^2 \\ y=\sqrt{\sqrt{x}-1},\:y=-\sqrt{\sqrt{x}-1},\:y=\sqrt{-\sqrt{x}-1},\:y=-\sqrt{-\sqrt{x}-1}$$