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?