I am interested on the functions $g:ℂ×ℂ→ℂ$ of the form $g(x+iy,x-iy)=g(z,\overline{z})$ I am interested on the functions $g:ℂ×ℂ→ℂ$ of the form $$g(x+iy,x-iy)=g(z,\overline{z})$$ My question is about requesting some references dealing with this type of functions.
 A: Depending on what class of maps $g$ you are wanting to consider, the answer to this question will vary greatly. I'm going to assume that since you've tagged this question as complex analysis, that you are assuming $g$ to be holomorphic.
First note that any holomorphic function $g\colon \mathbb{C}\times \mathbb{C}\to \mathbb{C}$ is real analytic, and so is the composition $z\mapsto g(z,\overline{z})$. Thus the class of maps you are interested in is contained in the class of real analytic maps $\mathbb{C}\to \mathbb{C}$.
On the other hand, suppose that $f\colon \mathbb{C}\to \mathbb{C}$ is real analytic. Then it can be written as a convergent power series $$f(x,y) = \sum_{n,m\geq 0}a_{nm}x^ny^m,$$ where $a_{nm}\in\mathbb{C}$. But note that you can rewrite this as a power series $$f(x,y) = \sum_{n,m\geq 0} a_{nm}\left(\frac{z+\overline{z}}{2}\right)^n\left(\frac{z-\overline{z}}{2i}\right)^m = \sum_{k,l\geq 0}b_{kl}z^k\overline{z}^l =g(z,\overline{z}),$$ where $g(z,w) = \sum_{k,l\geq 0}b_{kl}z^kw^l\colon \mathbb{C}\times\mathbb{C}\to \mathbb{C}$ is holomorphic.
So, unless I screwed this up (you should check, for instance, that the power series for $g$ converges!), the class of functions you want information about is just the class of real analytic functions $\mathbb{C}\to \mathbb{C}$.
For a reference on real analytic functions, see for instance the book A Primer of Real Analytic Functions by Krantz and Parks. Maybe other people have other suggestions.
