Covering map from sphere with six points removed to doubly-punctured complex plane 
$X$ is $S^2\subset \mathbf{R}^3$ with its intersection points with the coordinate axes removed.
Show that the following map is a covering map. $$\begin{align*}p:X&\longrightarrow \mathbf{C}-\{0,1\} \\ (x,y,z)&\longmapsto \left(\frac{x+iy}{1-z} \right)^4 \end{align*}$$

To check if it is a covering map, I should show that it is (1) continuous, (2) surjective and (3) that for all $w\in\mathbf{C}-\{0,1\}$ there is an open neighbourhood $U$ for which $p^{-1}(U)$ is the disjoint union of subsets $V_\alpha\subset X$ which are all homeomorphic to $U$.
We have never encountered such a difficult covering map before, the only one we have seen in class is the standard covering map $\mathbf{R}\to S^1:t\mapsto e^{it}$ and $S^1\to S^1:z\mapsto z^n$ for the circle $S^1$.
I know that the formula for stereographic projection $S^2-\{(0,0,1) \}\to \mathbf{C}$ from the north pole $(0,0,1)$ is $(x,y,z)\mapsto \frac{x+iy}{1-z}$. This is clearly a bijection $\phi:S^2-\setminus \{0,0,1\}\leftrightarrow\mathbf{C}-\{0\}$. Under $\phi$, the missing points of $X$ are mapped to $0,1,-1,i$ and $-i$. The map $z\mapsto z^4$ then 'rotates' the plane such that the points $1,-1,i,-i$ are sent to $1$.
The function is continuous since it's a formula, and $z\neq 1$ since we removed the point $(0,0,1)$ from the sphere.
I tried proving that it is surjective by using spherical coordinates, but that did not work out. Since $X$ does not contain $(0,0,0)$ we never reach $0$ in the image, but then why don't we reach $1$?
Could someone provide any help?

In the light of the comments to the question there is a more elegant answer.
The map $f : S^1 \to S^1, f(w) = w^4$, is known to be a covering projection with four sheets. Define $h : (0,\infty) \times S^1 \to \mathbb C^* = \mathbb C \setminus \{ 0 \} , h(t,w) = t w$ and $\phi : (0,\infty) \to (0,\infty), \phi(t) = t^4$. Both $h$ and $\phi$ are homeomorphisms. Define $F = h \circ (id \times \phi) \circ (f \times id) \circ h^{-1} : 
 \mathbb C^* \to \mathbb C^*$. Now $f \times id$ is a covering projection with four sheets, hence also $F$ is one. We have $F(h(t,w)) = (h \circ (id \times \phi) \circ (f \times id))(t,w) = h(w^4,t^4) = t^4w^4 = (tw)^4 = h(t,w)^4$, i.e.
$$F(w) = w^4.$$
The stereographic projection $s : S^2 \setminus \{(0,0,1)\} \to \mathbb C, S(x,y,z) = \dfrac{x+iy}{1-z},$ is a homeomorphism. It restricts to a homeomorphism $s^* : S^2 \setminus \{(0,0,1),(0,0,-1) \} \to \mathbb C^*$. Hence $P = F \circ s^*$ is a covering projection with four sheets and so is the restriction
$$P' : P^{-1} (\mathbb C^* \setminus \{ 1 \}) \to \mathbb C^* \setminus \{ 1 \} =  \mathbb C \setminus \{ 0, 1 \} .$$
But now $P^{-1} (\mathbb C^* \setminus \{ 1 \}) = \left( S^2 \setminus \{(0,0,1),(0,0,-1) \} \right) \setminus P^{-1}(1) = X$ and $P' = p$.
