Remove["Global`*"]

m = {{1, -a + b, 3}, {b, -6, 4}, {3, b, 0}};

VecMat[a_, b_] = Eigenvectors[m];

VecMatQ[a_, b_] = VecMat[a, b]*Conjugate[VecMat[a, b]];

PosVec[a_, b_] =
  Table[Position[VecMatQ[a, b][[i]], Max[VecMat[a, b][[i]]]], {i, 1, 3}] // Flatten;

PosVec[1, 1] // MatrixForm

Table[Position[VecMatQ[1, 1][[i]], Max[VecMat[1, 1][[i]]]], {i, 1, 3}] // Flatten