This is a short paper in which I use the twistor fibration of complex projective 3-space over the 4-sphere to construct, for each compact Riemann surface, a conformal and minimal immersion of that surface into the 4-sphere.
The idea is that complex projective 3-space has a natural, SO(5)-invariant holomorphic contact structure and, under the twistor fibration, holomorphic contact curves project conformally to minimal surfaces in the 4-sphere. As a holomorphic contact manifold, complex projective 3-space is birationally equivalent to the projectivized tangent bundle of the complex projective plane. Since every compact Riemann surface occurs as an immersed curve in the complex projective plane, it's just a matter of putting it in general position (so as to avoid the singularities of the birational transformation) in order to get every compact Riemann surface as an embedded contact curve in complex projective 3-space.
Not every minimal surface in the 4-sphere arises this way (although all of the genus 0 ones do), and I unfortunately coined the term super-minimal to refer to the ones that do. There are several reasons to abandon this terminology now and I discourage its use. It would be better if they were to be called isotropic, in consonance with the usage in the theory of harmonic maps.
Reprints are available.