Rotation around a whole sphere by multiplying a single hypercomplex number forever?

Question

In quaternion number system, any unit quaternion $\mathbf{q}\in\mathbb{H}$ can be written as $$ \mathbf{q} = \cos \theta + (v_x \mathbf{i} + v_y \mathbf{j} + v_z \mathbf{k})\sin \theta $$ for some $\theta \in [0,2\pi)$ and unit vector $v = (v_x,v_y,v_z)\in\mathbb{R}^3$. Since $v_x \mathbf{i} + v_y \mathbf{j} + v_z \mathbf{k}$ is a square root of $-1$, $\mathbf{q}^k$ is isomorphic to $(\cos\theta + \mathbf{i}\sin\theta)^k \in \mathbb{C}$ where $k\in\mathbb{Z}$. So if we start with $1$ and keep multiplying it by $\mathbf{q}$, we will end up rotating in a 2-d unit circle in a copy of $\mathbb{C}$ sitting inside $\mathbb{H}$, rather than rotating around a whole 4-d or 3-d sphere. So my question is, is it possible to rotate around a whole sphere with dimension at least 3, by keeping multiplying a single hypercomplex number in other generalized algebra like Clifford?