EDIT1:
How is it possible to find/solve for a continuously differentiable function of a complex variable $~f[x+iy]~$ if its real and imaginary parts satisfy:
$$x^{\Re[f]} + y^{\Im[f]} = x^2 + y^2~ $$where $~\Re[f]~:~$ real part of $~z~$ and $~\Im[f]~:~$ imaginary part of $~z~$?
Are there solutions other than complex constant $~2+2i~?$
EDIT2:
Stated in other words.. can we solve for two $(u,v)$ components satisfying Cauchy-Riemann conditions when total three equations (two among them partial differential, one algebraic ) are given:
$$ \frac{\partial u}{ \partial x }= \frac{\partial v}{ \partial y }, \frac{\partial u}{ \partial y }= -\frac{\partial v}{ \partial x } $$
$$x^u + y^v = x^2 + y^2? $$
What arbitrary functions can form a solution or part of it?
EDIT3:
Before being migrated from Mathematica, one response that produced a voluminous output but needed caution about relevance is:
FindInstance[x^rez + y^imz == x^2 + y^2, {rez, imz, x, y}, Reals, 20]
$$x^{u(x,y)}+y^{v(x,y)}=x^2+y^2$$ has innumerable solutions.
Choose $v(x,y)$ arbitrarily, and
$$u(x,y)=\log_x(x^2+y^2-y^{v(x,y)}).$$
Computing the partial derivatives
$$ux^{u-1}+\log x\,x^u\,u_x+\log y\,y^v\,v_x=2x$$
$$uy^{u-1}+\log x\,x^u\,u_y+\log y\,y^v\,v_y=2y$$
and applying Cauchy-Riemann,
$$\log x\,x^u\,u_x-\log y\,y^v\,u_y=2x-ux^{u-1}$$
$$\log y\,y^v\,u_x+\log x\,x^u\,u_y=2y-uy^{u-1}.$$
So we can get $u_x,u_y$. Not sure this leads us anywhere...