Which of the following sets are compact:
$1$ is closed and bounded and hence compact,$2$ is closed but not bounded and hence not compact.
$3$ is compact by Tychonoff Theorem and $4$ is not bounded and hence not compact.
Are these correct?
That’s correct. However, 1, 2 and 4 need a proof.
All three sets are closed, being inverse images of a closed set under a continuous function.
The set in 1 is bounded, because it is contained in $[-1,1]^3$.
The sets in 2 and 4 are not bounded, because they contain element with arbitrarily large norm; can you show them?
Easy: you can take $z=a+bi$ with arbitrary $b$.
Consider $z_3=1$. Then you can take $z_2=iz_1$, for arbitrary $z_1$.