Is it possible to operate on a single qubit by a map which has a degree not equal to one?

Let $c=c_0|0\rangle + c_1|1\rangle$ represent a qubit state where $c_0,c_1 \in \mathbb{C}$ and $|c_0|^2+|c_1|^2=1$. Then $c$ naturally lives in a 3-sphere $S^3$.

There is also the notion that $c$ lives in a one dimensional complex projective Hilbert space, $\mathcal{P}(\mathcal{V}_2) \cong \mathbb{CP}^1 \cong S^2$. In this case, we have the Hopf fibration $S^1 \hookrightarrow S^3\rightarrow S^2$. Generally, $\mathcal{P}(\mathcal{V}_N)\cong \mathbb{CP}^{N-1} \cong S^{2N-1}/U(1)$.

Single qubit operations are maps $f:S^3 \rightarrow S^3$ or $g:\mathbb{CP}^1\rightarrow \mathbb{CP}^1$. It is natural to wonder if $f$ or $g$ can be not homotopic to the identity. That is $\deg(f) \ne 1$ or $\deg(g)\ne 1$, given that $\pi_3(S^3) \cong \mathbb{Z}$ and $\pi_2(S^2) \cong \mathbb{Z}$.

Usually, a single qubit gate is either $\bf{X}$, $\bf{Y}$, $\bf{Z}$, $\bf{H}$, $\bf{S}/\bf{P}$, $\bf{T}$ which are order 2, 4, and 8, respectively and rotate (Pauli), mix (Hadamard), or phase shift states. These must be homotopic to the identity because the unitary group is path connected.

One can imagine an invertible map $f_n:S^3\rightarrow S^3$ which maps each point $e^{i\theta}$ in the $S^1$ fiber of the Hopf bundle to $f_n(e^{i\theta})=e^{in\theta}$.

Is $f_n$ a valid qubit operation? One could generalize to $N$-qubit maps (algorithms).