Theoreticaly, there exist immersions of the real projective space $\mathbb{R}P_3$ in $\mathbb{R^4}$.
If an equation for one is given, eg $f:\mathbb{R}P_3 \rightarrow \mathbb{R^4}$ you can visualize it as the cross-sections of a hyperplane moving along the orthogonal dimension.
You set the coordinates as $(x,y,z,t)$, where $t$ is time, and imagine the set of $(x,y,z)$ such that $(x,y,z,t)$ is in the image of $f$ at the given $t$.
By applying this procedure for the embedding of the real projective plane $\mathbb{R}P_2$ (as was asked here), I've sketched the following example:
For projective space, I'm having trouble getting either a visualization or an equation (for an immersion). I know if I could "glue" a $3$-ball to the real projective plane then it would give $\mathbb{R}P_3$ . That would mean starting with a point that grows into a sphere instead of loop, but I can't apply the "twisting" without getting cone-like singularities. So that's not a good strategy. I know there's other ways to obtain $\mathbb{R}P_3$, for example gluing two solid tori, but it's hard translating the correct "gluing" to immersion.
Please provide an explicit function for immersion of $\mathbb{R}P_3$ in $\mathbb{R^4}$, or guidance on how to correctly pick the cross-sections in $\mathbb{R}^3$ along time .