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The level equation is $|z_1{|}^2+|z_2{|}^2=|z_0{|}^2$. In particular $z_0\ne 0$, so the global affine coordinates on this level are $w_j=z_j/z_0$. They identify it withThe diagonal circle action becomes $(w_1,w_2)\mapsto (t{w}_1,t{w}_2)$ and is free: at least one coordinate is nonzero, so fixing a point forces $t=1$. Since the circle is compact, the Free proper Lie-group action theorem makes the quotient a smooth two-dimensional smooth manifold.

The map $(w_1,w_2)\mapsto [w_1,w_2]$ is the Hopf fibration. It is onto the complex projective line, and two unit representatives determine the same complex line precisely when they differ by a scalar of modulus one, hence by a circle orbit. Local sections, for example $[1,u]\mapsto (1,u)/\sqrt{1+|u{|}^2}$, show that the induced identification is a diffeomorphism. ThereforeThis identifies the symplectic reduction as a smooth manifold; no computation of its reduced symplectic form is needed.