$\displaystyle u_x(x) = v_y(y) \implies u_x(x) = v_y(y) = \lambda $ for some $\displaystyle \lambda \in \mathbb{R}$, because $\displaystyle u_{xx} = \partial v_y(y) / \partial x = 0$.

Thus $\displaystyle u(x) = \lambda x + b_0, v(y) = \lambda y + b_1$, so $\displaystyle f(z) = f(x+iy) = u(x) + iv(y)$

$\displaystyle = (\lambda x + b_0) + i(\lambda y + b_1) = \lambda (x+iy) + (b_0 + ib_1) = \lambda z + c$, where $\displaystyle c=b_0 + ib_1$.

Thus $\displaystyle f(z) = \lambda z + c$ for some fixed $\displaystyle \lambda \in \mathbb{R}, c \in \mathbb{C}$.