= Solution
The level equation is $|z_1|^2+|z_2|^2=|z_0|^2$. In particular $z_0\ne0$, so the global affine coordinates on this level are $w_j=z_j/z_0$. They identify it with
$$
N=\{(w_1,w_2)\in\mathbb C^2:|w_1|^2+|w_2|^2=1\}=S^3.
$$
The diagonal <circle> action becomes $(w_1,w_2)\mapsto(tw_1,tw_2)$ 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>. Therefore
$$
\boxed{M_{\mathrm{red}}=\mu^{-1}(-1/4)/S^1\cong\mathbb{CP}^1\cong S^2.}
$$
This identifies the <symplectic reduction> as a <smooth manifold>; no computation of its reduced <symplectic form> is needed.
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