Second order differential equation of a complex function

First solve for $w$ and then separate $u$ and $v$ from it:

solw[z_] = w[z] /. First@DSolve[w''[z] - (2 π I) w[z] == 0, w[z], z]
{solu[z_], solv[z_]} = {Re@#, Im@#} &@solw@z // ComplexExpand

You can use ReIm instead of {Re@#, Im@#} & if you're in v10, of course.

I would use @xzczd's approach most of the time. But if someone wants to solve the ODE in terms of component functions u and v, how to do it might not be so obvious to all users. You need to separate real and imaginary parts and simplify it with ComplexExpand.

DSolve[
 ComplexExpand[ReIm[D[u[z] + I v[z], z] - 2 Pi I (u[z] + I v[z])], z] == {0, 0},
 {u, v}, z]
(*
{{u -> Function[{z}, C[1] Cos[2 π z] - C[2] Sin[2 π z]], 
  v -> Function[{z}, C[2] Cos[2 π z] + C[1] Sin[2 π z]]}}
*)