Find a function $f:\mathbb{CP}^1 \times \mathbb{CP}^1 \longrightarrow \mathbb{CP}^3$ with image
$$ \{[z_0:z_1:z_2:z_3] \in \mathbb{CP}^3 \mid z_0^2 + \cdots + z_3^2 = 0\}. $$
I have got no idea how to get the solution. I know it has to involve some imaginary $i$ because otherwise, when we plug in purely real input $([x_0:x_1],[y_0:y_1])$, it's not possible to obtain $z_0^2 + \cdots + z_3^2 = 0$.