Can one classify the conformal embeddings of the Riemann sphere into four dimensional Euclidean space?

Question

I am interested in conformally embedding the Riemann sphere into four dimensional Euclidean space. One can trivially embed the Riemann sphere into three dimensions by identifying the usual sphere with the Riemann sphere. I think it should be possible to also embed it as an ellipse. I am wondering if those embeddings are unique, and if not whether they have been classified. I have looked into it and have not found any papers looking into this.