Let $a = z^2, b = 4$. By the triangle inequality, $|a + b| \leq |a| + |b|$. Therefore, by substitution, $|z^2 + 4| < |z^2| + 4 < |z^2| - 4$, and note $|z^2| == |z|^2$.
To visualize the triangle inequality in the complex plane, the lengths of the two sides of the origin are given as $|a|, |b|$, and the third side is $|a + b|$. As expected, the third side is shorter than the sum of the two sides.
The complex triangle inequality can be proved by squaring.
