My apology in advance if my question is obvious or elementary

We identify elements of $S^3$ with their quaternion representation $ x_1+x_2 i +x_3 j +x_4 k$. There are two independent vector fields $S_1(a)=ja$ and $S_2(a)=ka$ on $S^3$. On the other hand $S^3\to S^2$ is a $S^1$-principal bundle with the obvious action of $S^1$ on $S^3$. Then the span of $S_1, S_2$ is the standard horizontal space associated to the standard connection of the principal bundle $S^3 \to S^2$. Then each horizontal space has an almost complex structure $J$. This is the standard structure associated to $S_1, S_2$ coordinate.

Is this structure invariant under the action of $S^1$? If yes, we can define a unique almost complex structure on $S^2$ which is $P$ related to the structure on total space. Now is this structure on $S^2$ integrable?

One can ask a similar and modified question for $S^{2n+1} \to \mathbb{C}P^{n}$. Does the Horizontal space admit an invariant almost complex structure?

As a similar question, is there an example of a principal bundle $P\to X,$ such that $P$ is a real manifold and $X$ is a complex manifold and a connection admit an invariant almost complex structure which project to a non integrable structure?