Ω={{0,0,1,-I}, {0,0,I,-1}, {1,-I,0,0},{I,-1,0,0}};
MatrixForm[Ω](* Observable *)

ES=Eigensystem[Ω]

V=ES[[2]](* Not necesarily orthonormal basis *)
MatrixForm[V]

VP=V[[{3,4}]] (* Basis for  Sqrt(2) eigenspace *)
(* MatrixForm[VP]*)
VM=V[[{1,2}]] (* Basis for -Sqrt(2) eigenspace *)

VPO=Orthogonalize[VP](* Orthonormal basis for  Sqrt(2) eigenspace *)
(*MatrixForm[VP]*)
VMO=Orthogonalize[VM](* Orthonormal basis for -Sqrt(2) eigenspace *)

ketw1=Transpose[{VPO[[1]]}];MatrixForm[ketw1]
ketw2=Transpose[{VPO[[2]]}]; MatrixForm[ketw2]
ketw3=Transpose[{VMO[[1]]}]; MatrixForm[ketw3]
ketw4=Transpose[{VMO[[2]]}];  MatrixForm[ketw4]

braw1=ketw1\[ConjugateTranspose];MatrixForm[braw1]
braw2=ketw2\[ConjugateTranspose];MatrixForm[braw2]
braw3=ketw3\[ConjugateTranspose];MatrixForm[braw3]
braw4=ketw4\[ConjugateTranspose];MatrixForm[braw4]

P1=ketw1.braw1; MatrixForm[P1]
P2=ketw2.braw2; MatrixForm[P2]
P3=ketw3.braw3; MatrixForm[P3]
P4=ketw4.braw4; MatrixForm[P4]

P1.P1==P1(*Check to see if P1 is really a projector*)
P2.P2==P2(*Check to see if P2 is really a projector*)
P3.P3==P3(*Check to see if P3 is really a projector*)
P4.P4==P4(*Check to see if P4 is really a projector*)

P1.P2==DiagonalMatrix[{0,0,0,0}] (*Verifying orthogonality*)

Ω==(√2)*P1+(√2)*P2+(-√2)*P3+(-√2)*P4 (*Checking*)

P1+P2+P3+P4==DiagonalMatrix[{1,1,1,1}]

PP=P1+P2; MatrixForm[PP] (* Projector for Sqrt(2) eigenspace*)
PM=P3+P4; MatrixForm[PM](* Projector for -Sqrt(2) eigenspace*)

PP.PP==PP(*Checing to see that PP is a projector*)
PM.PM==PM(*Checking to see that PM is a projector*)

ketΨ=Transpose[{Normalize[ ({1,I,0,-1} )] } ];MatrixForm[ketΨ]
(*We will measure this state wrt Ω*)
braΨ=ketΨ\[ConjugateTranspose];MatrixForm[braΨ]

p1=Expand[   (braΨ.PP.ketΨ)[[1]][[1]]   ](* Probability of reading the eigenvalue  √2*)
p2=Expand[   (braΨ.PM.ketΨ)[[1]][[1]]   ](* Probability of reading the eigenvalue -√2*)

p1+p2==1(*Check*)

ketΨP=Map[Simplify,Transpose[  {   Normalize[   Transpose[PP.ketΨ ][[1]]   ]   }  ]];MatrixForm[ketΨP](* Resulting state if egenvaue √2 is read*)
ketΨM=Map[Simplify,Transpose[  {   Normalize[   Transpose[PM.ketΨ ][[1]]   ]   }  ]];MatrixForm[ketΨM]
(* Resulting state if egenvaue -√2 is read*)