I am trying to prove that set of non-zero quaternions is path connected. It is $S^3$, but I am not using path connectedness of higher dimensional spheres (except $S^1$ and $S^2$). I am using the book of Artin (Algebra).
For this, given any non-zero quaternion $q$, we can bring it to unit quaternion by map $t\mapsto \frac{q}{\|q\|}t + (1-t)q$ where $t\in [0,1]$.
So from $q$, we can move by a path to unit quternion.
Next, since $q/\|q\|$ is some unit quaternion, it lies on longitude of $S^3$, which is a unit circle passing through $\pm (1,0,0,0)$ where we identify $a+bi+cj+dk$ with $(a,b,c,d)\in\mathbb{R}^4$.
Thus, every non-zero quaternion is path connected to $1$.
Is this proof correct?