This is an old qual question:
Is there a complex structure on $\mathbb{R}^2$ (i.e., an atlas of charts making $\mathbb{R}^2$ a complex analytic manifold) such that $f:\mathbb{R}\to\mathbb{C}$, $f(x,y) = x-iy$ is analytic?
I'm really not sure even how to get started. My intuition would be that there is not, but I don't know how to prove that no such structure would exist. Could someone please point me in the right direction?
Yes. For clarity, I will write $X= \mathbb{R}^2$ with the new complex structure. The coordinate chart $g:X\to \mathbb{C}$ is given by conjugation: $g(x+iy) = x-iy$. Since this coordinate chart covers all of $X$, we do not have to check holomorphicity of any transition maps since there are none.
To decide if the function $f:X\to \mathbb{C}$ is holomorphic, we need to check that $f\circ g^{-1}: \mathbb{C}\to \mathbb{C}$ is holomorphic. But $f\circ g^{-1}(z) = \bar{\bar{z}} = z$ is the identity map, which is certainly holomorphic. Hence, $f$ is holomorphic with this "conjugate" complex structure on $\mathbb{R}^2$.