Abstract deformations of the unit sphere in $\mathbb{C}^2$ are encoded by complex functions on the sphere $S^3$. In sharp contrast with the higher dimensional case, for deformations of $S^3$ the natural integrability condition is vacuous and generic deformations are not embeddable even in $\mathbb{C}^N$ for any $N$. It is therefore a very difficult problem to characterize when a complex function on $S^3$ gives rise to an actual deformation of $S^3$ inside $\mathbb{C}^2$. I will discuss some recent work in this direction; this is current joint work in progress with Peter Ebenfelt.