Vtyjkyuk

So the circle $S^1$ is a hypersurface of $mathbb C$ and hence a generic CR submanifold. But $mathbb Csetminus0$ is the complexification of $S^1$ and $mathbb Csetminus0subset mathbb C$. Can $S^1$ be also a generic CR submanifold of $mathbb Csetminus0$ ?

And in general, $mathbb Rtimes S^2$ is a generic submanifod of $mathbb C^2$, $(mathbb Csetminus0)^2$, or $mathbb Ctimes( mathbb Csetminus0)$?

Yes, $S^1$ is a generic submaifold of $mathbb C setminus 0 $ as being generic is a local condition, so it is a generic submanifold of any neighborhood of $S^1$ in $mathbb C$.

Similarly $mathbb R times S^2$ is going to be a generic submanifold in any complex dimension 2 mmanifold, since however you embed it, it will be a hypersurface, and a hypersurface is always generic.

Again, the main thing is that being generic is a local condition, that is, $M$ is generic at $p$ if $T_p M + J T_p M$ ($J$ is the complex structure) is the entire tangent space of the ambient manifold.