How to prove this polynomial inequality?

How can we prove the following?

If $\frac{dP_{n}}{dz}|_{z=z_{0}}=0$ then $|P_{n}(z_{0})|<2$ for all $n>1$, where $P_{n}(z)\equiv P_{n-1}^{2}+z$ and $P_{1}\equiv z$

$z$ is in the complex plane.

It appears that $\lim{}_{n\rightarrow\infty}\max\{|P_{n}(z_{0})|:P_{n}'(z_{0})=0\}=2
$, but all I really need is a proof that it’s always <2 not that it approaches 2.

Both are easy to demonstrate with the Mathematica code below, but I can’t figure out a proof.

p[n_, z_] := If[n > 1, p[n - 1, z]^2 + z, z];
Do[Print[Max[Abs[p[n, z] /. Solve[D[p[n, z], z] == 0, z]] // N]], {n, 2, 8}]

Output:

0.25

1.15268

1.76235

1.94118

1.98545

1.99638

1.9991

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