differential equations - Plotting NDSolve function in complex coordinates

I have this system of equations:

$- \ddot{z} + \frac{1}{g} \frac{\partial g}{\partial z} \dot{z}^2 + \frac{1}{g} \frac{\partial g}{\partial z^*} \dot{z} \dot{z}^*$ =0 ,

$- \ddot{z}^*+ \frac{1}{g^*} \frac{\partial g^*}{\partial z^*} \dot{z}^{*2} + \frac{1}{g^*} \frac{\partial g}{\partial z} \dot{z} \dot{z}^*$ =0 .

I have written them as:

eq1 := - D[z[t],t,t] + (1/g[t]) * (D[ g[t],t]/ D[ z[t],t] ) * D[z[t],t]^2 + (1/g[t]) * (D[ g[t],t]/ D[ Conjugate [z[t]],t] ) * \dot{z} \dot{z}^*

eq2 := - D[Conjugate[z[t]], t,t] + (1/Conjugate [g[t]] ) * (D[ Conjugate [g[t]], t]/ D[ Conjugate[z[t]], t] ) * D[Conjugate[z[t]],t]^2 + (1/Conjugate [g[t]] ) * (D[g[t], t]/ D[ z[t], t] ) * \dot{z} \dot{z}^*

(* With initial conditions ) ( Note that these are arbitrary values, could be changed so that NDSolve give the most suitable solutions *)

ic= { g[0]==1, z[0]==1, Derivative[1][z][0]==0}

(* Numerically solved in z and g *)

solz := NDSolveValue[ {eq1==0, eq2==0, ic},z , {t,0,50} ]

solg := NDSolveValue[ {eq1==0, eq2==0, ic},g, {t,0,50} ]

Where I have used: $ \frac{\partial g}{\partial z} = \frac{\partial g}{\partial t} / \frac{\partial z}{\partial t} $

The question now, it’s apparent that both g and z are complex, something like:

z= z1 +i z2 , g= g1 +i g2

I will take z1= z2 = solz

So can I plot z in complex coordinates as a function in its real (solz ) and complex ( I solz) parts ?

Something like:

ParametricPlot3D[ { zRe, zIm, solz[ zRe+ I zIm], {zRe, 0,10}, {zIm,0,10} ]

(* this dose not work for that case because it requires solz to be in the form like:

solz[w_] := w * Conjugate[w]

And the same thing for g.

Hope all the problem’s details are clear.