Split the complex variables N3,N4 into real and imaginary part and ComplexExpand it.
GG = {N3*Conjugate[N3]+ N4*Conjugate[N4] == 1, (2 N3 + (-I + Sqrt[2]) N4)/(2 Sqrt[3]) == 0};
GG2 = GG /. {N3 -> n31 + I n32, N4 -> n41 + I n42}
ceRe = ComplexExpand[Re[GG2 /. Equal -> Subtract],
TargetFunctions -> {Re, Im}] // Simplify
ceIm = ComplexExpand[Im[GG2 /. Equal -> Subtract],
TargetFunctions -> {Re, Im}
] // Simplify
You get only solutions for three of the four variable, one you can choose free within a given range.
Here choose n42 from -2/Sqrt[7] to 2/Sqrt[7] and look at the solutions for the other variables.
sol = Solve[Join[Thread[ceRe == 0], {ceIm[[2]] == 0}], {n31, n32, n42, n41},
Reals] // Simplify
Plot[Evaluate[{n31, n32, n41} /. sol[[1 ;; 2]]], {n42, -(2/Sqrt[7]),
2/Sqrt[7]}, PlotStyle -> {Red, Green, Blue
}]

Or choose n31 free in the range +-Sqrt[3/7]
sol2 = Solve[Join[Thread[ceRe == 0], {ceIm[[2]] == 0}], {n32, n41, n31, n42},
Reals] // Simplify
Plot[Evaluate[{n32, n41, n42} /. sol2[[1 ;; 2]]], {n31, -Sqrt[(3/7)],
Sqrt[3/7]}, PlotStyle -> {Green, Blue, Magenta
}]
FullSimplify[Sols]
I get on Mathematica 12.1.1.0:{N3 -> ConditionalExpression[-(1/2) (-I + Sqrt[2]) N4, And[ Or[Im[N4] + (Rational[4, 7] - Re[N4]^2)^Rational[1, 2] == 0, Im[N4] == (Rational[4, 7] - Re[N4]^2)^Rational[1, 2]], Inequality[(-2) 7^Rational[-1, 2], LessEqual, Re[N4], LessEqual, 2 7^Rational[-1, 2]]]]}
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