Positivity of a complex quadratic form

Question

This should have a very simple intuitive explanation but I am not able to quite get it right now. Consider $z_0,z_1,z_2,z_3 \in \mathbb{C}$ where $z_0, z_3 \ge 0$. Suppose that

$$ z_0 + \lambda z_1 + \bar{\lambda}z_2 + \lambda\bar{\lambda} z_3 \ge 0 \quad \forall \lambda \in \mathbb{C} \, ,$$

where the bar indicates complex conjugation.

Now, why is it immediately clear that $z_1 = \bar{z}_2$?

What am I missing?

Answer 1

Setting $\lambda=1\,$ yields $z_1 + z_2 \in \mathbb{R}$, hence $\text{Im}(z_1)=- \text{Im}(z_2)$.

Setting $\lambda=i\,$ yields $i(z_1 - z_2) \in \mathbb{R}$, hence $\text{Re}(z_1)= \text{Re}(z_2)$.

Therefore $z_1=\bar{z_2}$.