Here $z = a + bi$, with $a, b, \in \mathbb{R}$ and $||z||_p = \sqrt[p]{|a|^p + |b|^p}$.
With $p = 1$, this is just diamond (square rotated 45 degrees) of side=$\sqrt2$ centered at the origin.
With $p = 2$, this is a circle of radius $1$ centered at the origin.
As $p \to \infty$, this is a square of side=$2$ centered at the origin.
Is there a geometric interpretation for each integer $p$?
For each $p\ge 1$, we have $\|z\|_p=1$ so $|x|^p+|y|^p=1$. Geometrically, this is what happens as $p$ varies:
(In the limit as $p\to\infty$, the edges sharpen to get a square.)