Any holomorphic function $f : U\to \mathbb{C}$ from a domain $U\subset\mathbb{C}$ induces a real-analytic mapping $f(x+iy)=u(x,y)+iv(x,y)$ which as such induces by complexification of $x,y$ and $u,v$ a holomorphic mapping $f^* : U^* \to \mathbb{C}^2$ for some domain $U^*\subset \mathbb{C}^2$. Of course the construction can be iterated to produce mappings $f^{*\cdots *}$ in $2^n$ complex variables for any $n\in \mathbb N$.
I guess that such "ultraholomorphic" functions have a standard name in the litterature, perhaps related to terms like Cauchy-Riemann but I wasn't able to find anything in my search engine. Is there some classic reference I should be aware of? Is there some nice/surprising property of $f^*$ (except the obvious real+Cauchy-Riemann)?
Given $R$ a subring of $M_n(\Bbb{C})$ let $$\eta(R)=\left\{\pmatrix{A&B\\-B&A},A,B\in R\right\}\subset M_{2n}(\Bbb{C}).$$ $\eta(R)$ is isomorphic to $R[t]/(t^2+1)$ and $\eta^2(R)$ is isomorphic to $R[t,u]/(t^2+1,u^2+1)$ and so on.
For $f(x)=\sum_{k\ge 0} c_k x^k$ an entire power series with real coefficients then you are just looking at $$f^{\overbrace{*\cdots *}^n} : \eta^n(\Bbb{C})\to \eta^n(\Bbb{C}), \qquad f^{\overbrace{*\cdots *}^n} (\alpha)= \sum_{k\ge 0} c_k \alpha^k$$ For $f(x)=\sum_{k\ge 0} (c_k+id_k) x^k$ then you are looking at $$f^{\overbrace{*\cdots *}^n} (\alpha)= \sum_{k\ge 0} (c_k+t d_k) \alpha^k$$ where $t$ is the $2^{n-1}\times 2^{n-1}$ diagonal matrix of $M_{2^{n-1}}(\eta(\Bbb{C}))$ where each diagonal entry is $\pmatrix{0&1\\-1&0} \in \eta(\Bbb{C})$.
Then your analytic function $\Bbb{C}^{2^n}\to \Bbb{C}^{2^n}$ is recovered from the natural vector space isomorphism $\Bbb{C}^{2^n}\to \eta^n(\Bbb{C})$.