Is the set $\mathbb C^2 \supset\{ y^2=x^2(1+x) \} $ a good example of irreducible analytic set, which does not define irreducible analytic germs at all of its points? I tried to construc it with some geometric insight (looking at what happens near the origin), now I'd like to prove the correctness (or not) of the statement
Conversely, as an example of analytic set which is not irreducible, but defines irreducible analytic germs at all of its points, can I take two disjoint planes in $\mathbb C^2$?
This is taken from exercise 1.1.11 in the book Complex geometry by Huybrechts
The answer to the first question is yes. Here is a hint: locally around the point $(0,0)$, the polynomial $1+x$ has a square root which can be written as a power series.
The answer to the second question is also yes: each $\mathbb{C}^2$ is an irreducible component and at any point the analytic germ does not see the other copy.