**NB: I have edited this question to clarify what the OP is asking – Robert Bryant**

Problem: Find a holomorphic function $f$ where where $f(x+iy) = u(x,y) + i\,v(x,y)$, such that the graph $\Gamma_u = \left\{\bigl(x,y,u(x,y)\bigr)\mid (x,y)\in\mathrm{dom}(f)\right\}\subset\mathbb{R}^3$ has Gauss curvature $K=-1$ at every point.

Because the Gauss curvature of such a graph $\Gamma_u$ is given by the formula $$ K(x,y) = \frac{u_{xx}u_{yy} - u_{xy}^2}{ (1 + u_x^2 + u_y^2)^2}, $$ the equation $K(x,y) = -1$ is a PDE of Monge-Ampere type. However, in this case, because $u$ is the real part of a holomorphic function and hence is harmonic, the function $u$ must also satisfy $u_{xx}+u_{yy} = 0$.

One wants to solve this combined system or else find a way to reduce it to an ODE so that one can investigate numerical solutions.