The problem is :
Show that the function $f(z)=2xy+i(x^2-y^2)$ is nowhere analytic
As it turns out the expression on the RHS is $i\bar z^2$. Then how come the book writes it as $f(z)$? If it were a function of $z$, we should have an expression for it in terms of $z$. But I can't seem to find such an expression.
To be totally strict (indeed, needlessly pedantic), the problem would be posed as:
where $z: \mathbb{R}^2 \to \mathbb{C}$ is defined by $z(x,y) := x + i y$.
Then, for instance, the conjugate function is $z \mapsto \bar{z}$, or strictly $z(x, y) \mapsto \overline{z(x,y)}$, given by $x + i y \mapsto x - i y$.
We may legally define a function $\mathbb{C} \to \mathbb{C}$ by defining its values on the image of $z$, because $z$ is bijective.