Applying the context-free pumping lemma to a language with crossed nestings
Pumping lemma lets you construct new strings belonging to the context-free language.

To prove that the language is not context-free you can take some string $s$ from L, use pumping lemma to construct a new string $s1$ which must belong to $L$ (because you used pumping lemma to construct it!). Then you show that $s1$ does not belong to L => contradiction.

Choice of original $s$ is yours. So yes, you can use $z = a^pb^pc^pd^p$ in your proof.