I'm studying analysis in several complex variables and in particular Weierstrass preparation theorem caught my interest (I'll include the theorem for clarity).
In the examples I came up with I only found functions that could be written as the product of an unit and linear Weierstrass polynomials.
It's never implied that every Weierstrass irreducible polynomial is of degree 1 or that in general irreducible functions are the degree 1 polynomial so I was looking for some examples of holomoprhic functions that are irreducible (at the origin) but are not polynomials of degree 1.
I actaully don't remember about details on why this is but I've been told in several occasions that polynomials like $z^2-xy^2$ are irreducible. I hope this helps