Stereographic projection $S^3 \to \mathbb{P}^2(\mathbb{C})$

Question

I think I can find a stereographic projection $S^2\setminus\{(0,0,1)\} \to \mathbb{P}^1(\mathbb{C})\setminus\{[0,1]\}$ using spherical coordinates:

it should be something like this $$(\theta,\phi)\to [1,2\tan (\frac \pi 4 + \frac \theta 2)e^{i\phi}]$$

Now I'm trying to find a projection $S^3 \setminus {(0,0,0,1)} \to \mathbb{P}^2(\mathbb{C}) \setminus \{Re(z_1=0)\}$

How can you express these projections in cartesian coordinates?

thank you