When we view $S^3$ as a subset of $\mathbb{C}^2$ through the equation $|z_1|^2+|z_2|^2=1$, the Hopf fibration $S^1\to S^3\xrightarrow{p} S^2$ is usually given by (see, e.g., Wikipedia) $$ p(z_1,z_2)=(z_1z_2^*,|z_1|^2-|z_2|^2). $$ The Hopf fiber can be identified with the $U(1)$ acting as $$ (z_1,z_2)\to (e^{i\phi}z_1,e^{i\phi}z_2). $$ I am interested in considering a different $U(1)$ which is generated by $$ (z_1,z_2)\to (e^{in\phi}z_1,e^{i\tilde{n}\phi}z_2) $$ for $n,\tilde{n}\in \mathbb{Z}$ general. When $\tilde{n}=n$ this reduces to the usual Hopf fiber, but in general now the base $S^2$ rotates as one moves along the associated fiber. Therefore, this version of $S^3$ is not anymore a fiber bundle in the sense that locally it is not described by a product of $S^2\times S^1$.
Is it correct to view this version of $S^3$ rather as an $S^2$ bundle over $S^1$? Any other perspective of this geometry would be appreciated.
The resulting spaces are known as lens spaces. These are an important source of counterexamples in low-dimensional topology. The homeomorphism classification is more subtle, and is given by Reidemeister torsion. Similar constructions are possible on all odd-dimensional spheres.