`NSolve` does the job by

    NSolve[E^(4 - 2 x - 2 E^(2 - 2 x) x) x - x == 0 &&  Abs[x] <= 10, x, Complexes];Dimensions[%]

> `{279, 1}`

Addition. It's strange that

    NSolve[E^(4 - 2 x - 2 E^(2 - 2 x) x) x - x == 0 && Abs[x] <= 10, x,  Complexes, WorkingPrecision -> 30];Dimesions[%]

> `{268, 1}`

and

    NSolve[E^(4 - 2 x - 2 E^(2 - 2 x) x) x - x == 0 && Re[x] <= 10 && 
    Re[x] >= -10 && Im[x] <= 10 && Im[x] >= -10, x, Complexes, 
    WorkingPrecision -> 30];Dimensions[%]

> `{268, 1}`

I'll try to verify it with Maple.