Why not $f(z)=z^2$ conformal at $z=0$?

Question

$$f(z)=z^2$$ is not conformal at $z=0$

Why?

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Conformal definition:

$f$ is conformal at z if f preserves angles there.

Answer 1

The angle between the rays $t$ and $it$ with $t\in[0,\infty)$ is $90^\circ$. The angle between thier images $t^2$ and $-t^2$ is $180^\circ$

Answer 2

For $f$ to be conformal $f'(z)\ne 0$. In your case $f'(z)|_{z=0}=0$.