 Nonvanishing tangent field on an odd-dimensional sphere

Identify ${\mathbb{R}}^{2m+2}$ with ${\mathbb{C}}^{m+1}$ and let $J$ be multiplication by $i$. On the unit sphere $S^{2m+1}$, the vector field $V(x)=Jx$ has norm one and satisfies $⟨x,Jx⟩=0$. It is therefore a continuous tangent nowhere-zero section, including on $S^1$.