Step 1:
$x^4+4=0$
$x^4=-4$
$\Rightarrow x^4=4(\cos\pi+i\sin\pi)$
Therefore $x=4^{\large\frac{1}{4}}(\cos\pi+i\sin\pi)^{\large\frac{1}{4}}$
$\qquad\qquad=4^{\large\frac{1}{4}}(\cos(2k\pi+\pi)+i\sin(2k\pi+\pi))^{\large\frac{1}{4}}\;\;\;k\in z$
$\qquad\qquad=4^{\large\frac{1}{4}}(\cos(2k+1)\pi+i\sin(2k+1)\pi)^{\large\frac{1}{4}}\;\;\;k\in z$
$\qquad\qquad=\sqrt 2(\cos(2k+1)\large\frac{\pi}{4}$$+i\sin(2k+1)\large\frac{\pi}{4})$$\;\;\;k=0,1,2,3$
Step 2:
The roots are $\sqrt 2 cis\large\frac{\pi}{4},$$\sqrt 2 cis\large\frac{3\pi}{4},$$\sqrt 2 cis\large\frac{3\pi}{4},$$\sqrt 2 cis\large\frac{5\pi}{4},$$\sqrt 2 cis\large\frac{7\pi}{4}$
$\Rightarrow \sqrt 2 cis\large\frac{\pi}{4},$$\sqrt 2 cis\large\frac{3\pi}{4},$$\sqrt 2 cis\large\frac{3\pi}{4},$$\sqrt 2 cis(\large\frac{5\pi}{4}$$-2\pi),\sqrt 2 cis(\large\frac{7\pi}{4}$$-2\pi)$
$\Rightarrow \sqrt 2 cis\large\frac{\pi}{4},$$\sqrt 2 cis\large\frac{3\pi}{4},$$\sqrt 2 cis\large\frac{3\pi}{4},$$\sqrt 2 cis(\large\frac{-3\pi}{4}),$$\sqrt 2 cis(\large\frac{-\pi}{4})$
$\Rightarrow \sqrt 2 cis\big(\large\frac{\pm \pi}{4}\big),$$\sqrt 2cis\big(\large\frac{\pm 3\pi}{4}\big)$