Germ surface of the square root of z to the eighth minus one
= Germ surface of the square root of z to the eighth minus one
Over $D=\mathbb C\setminus\{z:z^8=1\}$, the two-valued square root is represented by
$$
R=\{(z,w)\in D\times\mathbb C:w^2=z^8-1\}.
$$
The maps $\pi(z,w)=z$ and $\mathcal E(z,w)=w$ identify this unbranched double cover analytically with the corresponding component of the space of germs.