at.algebraic topology - Hopf fibration inside the retraction of R^4 minus line -> S^2?

First, note that $mathbb R^4 setminus mathbb R simeq S^2times mathbb R^2$.
So you are asking for an immersion $S^3to S^2times mathbb R^2$
representing the generator $eta$ of $pi_3(S^2times mathbb R^2)=pi_3(S^2)=mathbb Z$.

I'm guessing that your immersion doens't exist, and that you need to consider maps $S^3to S^2times mathbb R^n$ with larger $n$, in order to represent $eta$ by an immersion.

But if you are willing to go a little bit up in dimension, and consider maps $S^3to S^2times mathbb R^4$, then you can even find an embedding representing $eta$.
It is given by $(H,I):S^3to S^2times mathbb R^4$, where $H:S^3to S^2$ is the Hopf map, and $I: S^3 rightarrow {mathbb R}^4$ is the standard inclusion.