Bounded subsets of $\Bbb C^2$.

Question

Which of the following subsets of $\Bbb C^2$ are bounded?

1. $\{(z,w): z^2 + w^2 = 1\}$,

2. $\{(z,w): |Re z|^2 + |Re w|^2 = 1\}$,

3. $\{(z,w): |z|^2 + |w|^2 = 1\}$,

4. $\{(z,w): |z|^2 - |w|^2 = 1\}$.

For option $1$, I found some elements in the set but all have norm $1$. But still, I am not convinced.

option $3$ is the unit ball in $\Bbb R^4$ hence it is bounded. Other options I am not sure.

Please share your thoughts. Thank you.

Answer 1

1. It is unbounded, since it contains every pair of the form $(z,w)$, where $w$ is a square root of $1-z^2$.
2. It is unbounded, since it contains every pair $(1,it)$, with $t\in\Bbb R$.
3. You are right here.
4. It is unbounded, since it contains every par $(\cosh t,\sinh t)$, with $t\in\Bbb R$.