So the circle $S^1$ is a hypersurface of $\mathbb C$ and hence a generic CR submanifold. But $\mathbb C\setminus\{0\}$ is the complexification of $S^1$ and $\mathbb C\setminus\{0\}\subset \mathbb C$. Can $S^1$ be also a generic CR submanifold of $\mathbb C\setminus\{0\}$ ?
And in general, $\mathbb R\times S^2$ is a generic submanifod of $\mathbb C^2$, $(\mathbb C\setminus\{0\})^2$, or $\mathbb C\times( \mathbb C\setminus\{0\})$?
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.