好像除了换元+实虚分开+ComplexExpand以外没别的方法：
complexSolve[eqn_, var_] :=
Module[{a, b},
var -> With[{var = a + b I},
var /. Solve[{Re@
#, Im@#
} & /@ eqn // ComplexExpand, {a, b}]] // Thread]
complexSolve[z + Abs@z == 2 + I, z]
complexSolve[(1 + 2 I) Conjugate@z == 4 + 3 I, z]
如果有版本10的话，用ReIm能让代码简化点。

用Reduce...
Reduce[z + Abs[z] == 2 + I, z]
Reduce[(1 + 2 I) Conjugate[z] == 4 + 3 I, z]