Does there exist a function $f:\mathbb{C}^2\to \mathbb{C}$ holomorphic aside from at singularities with isolated singularities?

Question

All the examples I've been able to come up with are either non-holomorphic or fail the condition. For example, with the 1D case we can take $$f(z) = \frac{1}{1+z^2}$$ With singularities at $\pm i$. If we try to extend this to some $$f(z,w) = \frac{1}{1+(z+w)^2}$$We now have singularities at $z=-w\pm i$ Which is not at all what we wanted.

Answer 1

Hartogs extension principle (simple version using Hartogs figure suffices) means that there are no isolated singularities. There aren't even any compact singularities. If a function has an actual nonremovable "singularity", then that singularity will go off to infinity somehow, just like in your example.