Fiber bundle of sphere over projective space

Statement of general version
Suppose $$k = \mathbb{R}, \mathbb{C}, \mathbb{H}$$, i.e., $$k$$ is either the real numbers, or the complex numbers, or the Hamiltonian quaternions. Let $$| \cdot |$$ denote the absolute value/modulus operation in $$k$$. For $$x = (x_1,x_2, \dots, x_n)$$ an element of $$k^n$$, we define:

$$\! \| x \| = \sum_{t=1}^n |x_t|^2$$

Now define the sphere:

$$S^n(k) = \{ x \in k^{n+1} \mid \| x \| = 1 \}$$

with the subspace topology from the topology on $$k^{n+1}$$ arising from the product topology on $$k^{n+1}$$ from the usual Euclidean topology on $$k$$.

Note that $$S^0(k)$$ is a group, because it is the kernel of the modulus homomorphism from $$k \setminus \{ 0 \}$$ to the multiplicative group of nonzero reals.

Define the defining ingredient::projective space:

$$\mathbb{P}^n(k) = (k^{n+1} \setminus \{ 0 \})/(k \setminus \{ 0 \})$$

where the quotient is via the diagonal left multiplication action. We put the quotient topology from the subspace topology on $$k^{n+1} \setminus \{ 0 \}$$ arising from the product topology on $$k^{n+1}$$.

There is a fiber bundle $$S^n(k) \to \mathbb{P}^n(k)$$ with fiber $$S^0(k)$$. The map composes the inclusion of $$S^n(k)$$ in $$k^{n+1}\setminus \{ 0 \}$$ with the quotient map to $$\mathbb{P}^n(k)$$.

Interpretation in the three special cases
Interpretation for arbitrary $$n$$:

Interpretation for $$n = 1$$: In this case, $$\mathbb{P}^n(k)$$ itself becomes a sphere. We get some very special fiber bundles:

In fact, these are the only fibrations where the base space, total space, and fiber space are all spheres.