Let us define the numerical range for some matrix $A \in \mathbb{C}^{n \times n}$:
$W(A) := \lbrace x^\star A x \mid x \in \mathbb{C}^n, x^\star x = 1 \rbrace $
Prove that the same set can be obtained by only considering vectors whose $i$-th (for some fixed $i$ that satisfies $1 \leq i \leq n$) coordinate is positive (and thus real).
Presumably you mean "non-negative" as opposed to positive. Otherwise, the matrix $$ A = \pmatrix{1&0\\0&0} $$ is a counterexample.
With that said, it suffices to note that for every $x$ for which the $i$th entry $x_i$ fails to be non-negative, $y = x \cdot \frac{|x_i|}{x_i}$ has a positive $i$th entry and satisfies $y^*Ay = x^*Ax$.