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Can we have a function of a complex variable whose real and imaginary parts cannot be represented in cartesian Mongé $ z= F(x,y) $ form, but can only be represented as a parametrized surface $$ (x,y,z)= f(u,v), g(u,v),h(u,v))\quad ? $$

If not possible, why not? and if it is possible, please give some illustrative examples of such non-orthogonal parameterized cases.

For example for $ w=z^3$ the monkey saddle parts are both of Cartesian Mongé form only.

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    What do you mean "real and imaginary parts cannot be represented in Monge form"? – 2017-02-12
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    What is the connection here between the function and the surface? Are you looking at a real valued function? Then real and imaginary part makes no sense. If you are taking about a complex valued function it does not define a surface as a graph in $\mathbb{R}^3$. To me it's not clear what you are asking. – 2017-02-12
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    E.g., $. w=z^2, Re[w]= (x^2-y^2) , Im[w] =2 x y $ are hyperbolic paraboloid surfaces in Monge form. But Monge form is not the only way to represent a surface.. – 2017-02-12

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