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### Solving a differential equation with a complex (independent) varible

I'm trying to plug in complex values in the numerical solutions of ODEs without success. For instance y[I] /. NDSolve[{y'[x] == 0, y == 3}, y, {x, -10, 10}] ...
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### Instructing DSolve not to return answers with explicit complex numbers

DSolve sometimes returns solutions containing explicit complex numbers, even though the ODE does not. For instance (drawn from question 126072), ...
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### Differential equations with a complex variable

Is Mathematica able to handle ordinary differential equations where the variable itself is complex? I am looking for solutions of ODE systems of the form \left\{ \begin{align} i\frac{da_1}{dt} &=...
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^*} ...