Here's what I'm thinking about: if a function has a nonzero derivative it's dependant on its input variable. Likewise the contrapositive: if it's independent of the variable it's derivative with respect to that variable will be zero.
If $f$ is a function dependant on $z$, which can be given by its conjugate. Therefore $f$ is a function of the conjugate of its input, and should have a nonzero derivative.
Where did this thought process go wrong? I've seen this derivative referenced in a proof of the Cauchy Riemann equations and elsewhere.
EDIT: let f be complex differentiable
essentially the reason is that because $x$ and $y$ are independent variables, it follows that $z$ and $\bar{z}$ are also independent variables as there is a clear bijective transformation between them. So any "complex" function regarded as real in two variables $x$ and $y$ becomes then a function in the new variables $z$ and $\bar{z}$ and then you can define properties expressed in terms of these new variables, and those turned to be extremely useful like analyticity of a function meaning there is no $\bar{z}$ "involved" in the function (so analogous to having a function of only $x$ but not of $y$), harmonicity meaning that $z$ and $\bar{z}$ are "separated" but not mixed together etc - introducing the usual differential operators makes this rigorous but I think the intuitive meaning is clear.