More generally, consider the singularity given by $$x_1^2+\cdots+x_{n+1}^2=0$$ in $\mathbf{C}^{n+1}$. (Your case is $n=2$ after a change of variables.) Identifying $\mathbf{C}^{n+1}=\mathbf{R}^{n+1}\times\mathbf{R}^{n+1}$ we see at once that the link is the Stiefel manifold $\mathrm{V}_2(\mathbf{R}^{n+1})$ of pairs of orthonormal vectors in $\mathbf{R}^{n+1}$. If you allow more complicated exponents the link can be very interesting, for instance it may be an exotic sphere (see for example E. Brieskorn's classic paper Beispiele zur Differentialtopologie von Singularitäten).

More generally, consider the singularity given by $$x_1^2+\cdots+x_{n+1}^2=0$$ in $\mathbf{C}^{n+1}$. (Your case is $n=2$ after a change of variables.) Identifying $\mathbf{C}^{n+1}=\mathbf{R}^{n+1}\times\mathbf{R}^{n+1}$ we see at once that the link is the Stiefel manifold $\mathrm{V}_2(\mathbf{R}^{n+1})$ of pairs of orthonormal vectors in $\mathbf{R}^{n+1}$. (For $n=2$ this is $\mathrm{SO}(3)$, which is homeomorphic to $\mathbf{RP}^3$.) If you allow more complicated exponents in the defining equation of the singularity, the link can be very interesting. For example, the links of the singularities defined by $$x_1^2+x_2^2+x_3^2+x_4^3+x_5^{6k-1}=0 \ \ \ (1\leqslant k\leqslant 28)$$ give all 28 differentiable structures on $S^7$. (See for example E. Brieskorn's classic paper Beispiele zur Differentialtopologie von Singularitäten).