Let $H:=\lbrace y_1=y_2=0\rbrace$, where $z_j = x_j +\sqrt{-1}y_j$, $j = 1; 2; 3$ be a four dimensional subspace of $\mathbb{C^3}$.
Let $f:\mathbb{C}\rightarrow\mathbb{C^3}$ be holomorphic curve not identically zero ($f\neq 0$). Let $\pi:\mathbb{C^3}\setminus\lbrace0\rbrace\rightarrow \mathbb{C} P^2$ the canonical projection to the complex projective space $\mathbb{C} P^2$.
Question: if $f$ avoids $H$, must $\pi(f)$ avoid $\pi(H)$?.
Note: $\pi(H)$ is given by the closure of $\lbrace [1,Z_1,Z_2]\mid Z_2 ~ \text{is a real multiple of} ~Z_1\rbrace$.
This is not true. Consider the curve $f:\Bbb C\to \Bbb C^3$ given by $z\mapsto (z,i,i)$. Clearly this avoids $H$, but includes the point $(i,i,i)$ which maps to the same point as $(1,1,1)$ under $\pi$.
I would also caution you that the map $\pi$ behaves strangely on sets which are not stable under the scaling action by $\Bbb C^*$ in $\Bbb C^3\setminus 0$. Perhaps you have a good reason for pursuing this, but if I ended up thinking about something like this I'd be curious about whether I was on the right track. If this is a case of the XY problem, you may be better served by asking about your original problem.