Evaluate $\int_{-i}^{i}|z| d z$ along different contours. Does $|z|$ have an antiderivative?
Progress:
If we say $z = x + iy$, then $f(z) = |z| = \sqrt{x^2 + y^2} + i*0$, so Cauchy-Riemann equations only satisfied at origin and it is not analytic. Therefore, we cannot apply Cauchy Theorem.
If we say $z = r*e^{i\theta}$, then $|z| = r$ and it looks like for each contour, we will get different result.
I don't know how to proceed after that.
So, basically showing that $|z|$ is not analytic is enough to say that it has no antiderivative (Is a function analytic iff it has antiderivative?).
And we can simply take 2 contours: $C_1 = \{|Im z|<1 | Re z = 0 \}$ and $C_2 = \{|z|=1 | Arg z \in (-\pi, \pi) \}$ to get different results.