Is it true that holomorphic functions are not conformal? By my understanding a holomorphic function is angle-preserving.
But if I plot the lines $t+i$ and $1+it$ under the map $x \mapsto x^2$ then the images meet at $90^o$, but if I map the lines under $x \mapsto x^3$ then they do not. What am I missing?
First of all it is worth recalling that a holomorphic function $f$ preserves the angle between curves at those points for which $f'(z)\neq 0$. This is in the sense that if $\gamma_1, \gamma_2 $ are curves through $z$ then the angle between $f\circ\gamma_1 $, $f\circ \gamma_2$ at $f(z)$ is the same as that between $\gamma_1, \gamma_2 $ at $z$.
You should note that this is indeed the case in your plot; the image under $z\mapsto z^3$ of the intersection point $1+i$ is $-2+2i$ and the curves do meet at $90^o$ at this point.
So what's going on with the other intersection? Well this arises because $z\mapsto z^3$ is not injective. The other intersection corresponds to different points on the original curves. So there is no reason they should meet at right angles.
N.B. The fact that for $z\mapsto z^2$, the images meet at right angles at $-2$ the is pretty much a coincidence.