Prove that $\frac{z^{1/4}}{z^2+4}$ is not holomorphic

Question

I am trying to prove that the function $f(z)= \frac{z^{1/4}}{z^2+4}$ is not holomorphic in $0$. I was thinking, is it okay to just say that if I look at it as a $\mathbb{R}^2 \to \mathbb{R}^2$ function, it is not differentiable because its partial derivatives does not exist? I mean, is it enough to just say that the function is not differentiable because $f(x)= \frac{x^{1/4}}{x^2+4}$ is not derivable in $0$ (and the same for $y$ of course)?