Setting aside the codomain issue (it should be $\mathbb C$ instead of $\mathbb R$ if we want nonconstant holomorphic functions), the answer is that $\lim_{h\to 0}\frac{f(z+h) - f(z)}{|h|}$ is not a suitable limit for definition of derivative. While the difference-quotient-limit formalism is a convenient tool in some cases, the deeper meaning of the derivative is linear approximation to $f$. The existence of derivative at $z$ means that $f(z+h)-f(z) \approx \ell(h)$ on small scales, where $\ell$ is a linear function of $h$. In the complex domain we can write $\ell(z) = Az$ for some complex number $A$. This number is called the derivative $f'(z)$. It can be calculated by dividing the relation $f(z+h)-f(z) \approx Ah$ by $h$ (which I decided not to make precise by including the error term) and then letting $h \to 0$: $A=\lim_{h\to 0}\frac{f(z+h) - f(z)}{h}$