The authors study the moduli space of conformal immersions, \(f: M \to S^4\), of a given Riemannian surface \(M\) into the \(4\)-sphere \(S^4\), up to Möbius equivalence. This study focuses on the \(2\)-torus case, \(M=T^2\), and the map \(f: T^2 \to S^4\) is a conformal immersion with degree zero normal bundle. The authors associate to such a conformal immersion \(f: T^2 \to S^4\), the moduli space of generalized Darboux transforms, which has the structure of a Riemannian surface, the spectral curve. This allows the authors to describe conformal immersions of \(T^2\) into \(S^4\) in terms of some algebro-geometric data.