Is there a simple example of a transitive vector field on the three sphere? I was just thinking about the transitive field on the torus and the possibility of defining a vector field without singularities on $S^3$ and this idea popped up.
Edit: I think my question was misunderstood. I am specifically asking about transitivity, or the existence of a dense orbit (flow). I know about the existence of a vector field without singularities (which is what every answer is mentioning), I just worded my initial post poorly. What I want to know is if it is possible to define a vector field on $S^3$ that has a dense orbit.

This is not straight forward. You can take equivalence classes of paths through any point $(z,w)\in S^3$, where $S^3=\{(z,w)\in \mathbb{C}^2\mid |z|^2+|w|^2=1\}$, with paths given by $l_{z,w}:(-\varepsilon,\varepsilon)\to S^3$, $l_{z,w}(t) = (ze^{it},we^{it})$.
