Start with the definition of analytic. The function of a complex variable $z$ is analytic at $z \in \mathbb C$ if it is differentiable at $z$, which means
$$\begin{align} \frac{f(z+h) - f(z)}{h} \tag 1 \end{align}$$
has a unique limit as $\lvert h \rvert \to 0$, denoted $f'(z)$. The limit has to exist regardless of how and in which direction $h$ approaches zero.
This is a strong requirement and requires $f(z)$ to satisfy the Cauchy Riemann equations.
These are obtained as follows: write $z=x+iy$ and $f(z) = u(x,y)+iv(x,y)$ and consider the complex derivative when $h = \delta x$ and $h = i\delta y$ for real $\delta x,\delta y$. If $f$ is required to be analytic at $z$ then both are must be the same, so we obtain,
$$\frac{\partial f}{\partial x} = f'(z) = -i \frac{\partial f}{\partial y}.$$
Now write this in terms of $u,v$ to obtain the Cauchy-Riemann equations,
$$ \begin{align}
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad
\frac{\partial v}{\partial x} = -\frac{\partial u}{\partial y}. \tag 2
\end{align}$$
When applied to $\lvert z \rvert^3$ these break down. We have $u(x,y) = (x^2+y^2)^{3/2} $ and $v(x,y) = 0$. It is not difficult to see then that $(2)$ will only be satisfied by exception, when $x = y = 0$. Thus $\lvert z \rvert^3$ cannot be analytic except at the single point $z = 0$.
I hope this is useful.
