Maximization of a complex function with real outputs

Question

Let $a$ and $b$ be functions from $\mathbb{C}$ to $\mathbb{R}$. Suppose that for some real $\alpha$ and $\beta$, there exist complex $z$ and $w$ such that $a(z) = \alpha$ and $b(w) = \beta$. Given that $x$ and $y$ are complex numbers such that $x + y = z + w$, what is the maximum possible value of $a(x) + b(y)$ in terms of $a,b,\alpha,\beta$?