Nonlinear equation in two unknowns with unique solution over the field of complex numbers We know that a linear equation with two unknowns can not have a unique solution. On the other hand a nonlinear equation in two unknowns such as $$a^2+b^2=0$$ has a unique solution $a=0$ and $b=0$ over the real numbers.  

But does there exist a nonlinear equation in two unknowns which has a unique solution in the field of complex numbers? 

Restrict the space of functions to the elementary functions: http://en.wikipedia.org/wiki/Elementary_function.
It can be any complex number. For example a special case of imaginary number as sum of two squares:

$ a = x + i y,  b = u + i v ,$ 

$  a^2 + b^2 =  ( x^2 - y^2 + u^2 - v^2) + 2 i (x y + u v) $ 

if $ ( a,b) = ( (1,1), (-1,-1)),$  then $ a^2 + b^2 = 4 i. $
It can be any complex number. For example a special case of imaginary number as sum of two squares:

$ a = x + i y,  b = u + i v ,$ 

$  a^2 + b^2 =  ( x^2 - y^2 + u^2 - v^2) + 2 i (x y + u v) $ 

if $ ( a,b) = ( (1,1), (-1,-1)),$  then $ a^2 + b^2 = 4 i. $