I am having problems reconciling two results that seem to be at odds with each other.
On the one hand, I find references to the fact that $$\partial_z \mathrm{Re}(f(z)) = \mathrm{Re}(\partial_zf(z))$$ and therefore $$\partial_z\mathrm{Re(z)} = 1$$ On the other hand, $$\mathrm{Re}(z) = \frac{z+z^*}{2}$$ and so $$\partial_z \frac{z+z^*}{2}= \frac12$$ The caveat is that in the second case I'm treating $z$ and $z^*$ as independent variables, which I understand is the correct thing to do when we derive non-holomorphic functions.
Where is my mistake?