Monge-Ampere type PDE 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.
 A: Now I understand the OP's question, which I would state this way:  
Is there a holomorphic function $f(x,y) = u(x,y) + \mathrm{i}\,v(x,y)$ on a domain in the $xy$-plane such that the induced metric on the graph of $u$ has Gauss curvature $-1$?  (Note:  The OP needs $f$ to be holomorphic, not just complex, if the given formula for the Gauss curvature is going to be correct.)
The answer to this question is 'no'; here is why:  The hypotheses can be expressed in terms of $u$ as follows:  Find the common (local) solutions to the pair of equations
$$
u_{xx} + u_{yy} = 0\qquad\text{and}\qquad 
{u_{xx}}^2 + {u_{xy}}^2 = \bigl(1 + {u_x}^2+{u_y}^2\bigr)^2.
$$
I claim that there are no such solutions.  To see this, note that for any solution of these equations on a simply connected domain in the $xy$-plane, there must exist a function $\theta$ on that domain such that
$$
u_{xx} = \cos\theta\,\bigl(1 + {u_x}^2+{u_y}^2\bigr),\quad
u_{xy} = \sin\theta\,\bigl(1 + {u_x}^2+{u_y}^2\bigr),\quad
u_{yy} = -\cos\theta\,\bigl(1 + {u_x}^2+{u_y}^2\bigr).
$$
Differentiating these equations and expanding out $(u_{xx})_y = (u_{xy})_x$ and $(u_{xy})_y = (u_{yy})_x$ then shows that the function $\theta$ must satisfy the differential equations
$$
\theta_x = 2(u_x\,\sin\theta -u_y\,\cos\theta),\quad\text{and}\quad
\theta_y = -2(u_y\,\sin\theta +u_x\,\cos\theta).
$$
Finally, using these formulae and the above formulae for $u_{xx}, u_{xy}, u_{yy}$ to expand out the identity $(\theta_x)_y-(\theta_y)_x = 0$ then yields $4=0$, a contradiction.
Thus, there are no solutions of this kind.
A: There are some results in the literature reducing some Monge-Ampre and Gauss curvature type equations (and others) into an 'ODE'; see A NOTE ON SURFACES WITH RADIALLY SYMMETRIC NONPOSITIVE GAUSSIAN CURVATURE and the related article HOLOMORPHIC METHODS IN PDE 
