dg.differential geometry - Diffeomorphism group of the unit sphere of complex n-space

I think that you are asking for the group of automorphisms of the CR-structure on $mathbb{S}^{2n-1}$ as hypersurface in $mathbb{C}^n$ (assuming $n>1$, since $n=1$ is another story). Even locally, any such automorphism is induced by an element of $PU(n,1)$, which is the subgroup of $PGL(n+1,mathbb{C})=mathrm{Aut}(mathbb{C}P^n)$ preserving the unit ball $mathbb{B}subsetmathbb{C}^nsubsetmathbb{C}P^n$. This result is due to S.S. Chern and J. Moser, Acta Math. 133 (1974), 219--271 (but the global version you are asking for is perhaps older).

Concretely, $PU(n,1)$ is the isometry group of the hermitian form $|z_1|^2+dots+|z_n|^2-|z_{n+1}|^2$, quotiented by its scalar subgroup $U(1)$.
The diffeomorphism of $mathbb{B}subsetmathbb{C}^n$ associated to a matrix $Ain U(n,1)$ is given by $vmapsto (a_{11}v+a_{12})/(a_{21}v+a_{22})$, where $a_{ij}$ are the blocks of $A$ for the obvious decomposition $mathbb{C}^{n+1}=mathbb{C}^noplusmathbb{C}$ (thus $a_{22}$ is scalar, for instance, and $a_{21}$ is an $1times n$ line matrix).

Hope this answers your question