If $X$ is a compact oriented surface in a 4-dimensional oriented manifold $M$, then the self-intersection number $X^2$ of $X$ is given by the integral over $X$ of the Euler class of the normal bundle. In the case of $CP^1$ embedded in $CP^2$, the normal bundle is isomorphic to the Hopf bundle, therefore $X^2$ can be obtained calculating the first Chern number of the Hopf fibration (or equivalently the Euler number of its realization).

It is possible to have circle bundles on $CP^1$ with higher Chern number by taking the quotient of the total space of the Hopf fibration by the action generated by $(z^1,z^2)\mapsto(z^1 \exp(i 2 \pi/k), z^2 \exp(i 2\pi/k ))$.

Are these bundles the normal bundles of some embedding of $CP^1$ in a 4-dimensional manifold? If yes, is it possible to describe the embedding explicitly? Is there a deeper relation between the Hopf bundle and the normal bundle of $CP^1$ embedded in $CP^2$ or do they just happen to be the same?