  It is shown that analytic conformal submersions of $S^3$ are given by
intersections of (not necessary closed) complex surfaces with a quadratic real
hyper-surface in $\mathbb{C}P^3.$ A new description of the space of circles in
the 3-sphere in terms of a natural bilinear form on the tangent sphere bundle
of $S^3$ is given. As an application it is shown that every conformal fibration
of $S^3$ by circles is the Hopf fibration up to conformal transformations.
