Take elements from a list based on two criteria

Select[list, Count[#, 1] == 1&]

{{-1, 1}, {1, -1}, {1, (1 + I)/Sqrt[2]}}

Also

Pick[list, Lookup[Counts /@ list, 1, 0], 1]

{{-1, 1}, {1, -1}, {1, (1 + I)/Sqrt[2]}}

And alternative ways to use Cases:

Cases[{OrderlessPatternSequence[1, Except[1]]}]@list
Cases[_?(Counts[#][1] == 1&)]@list

{{-1, 1}, {1, -1}, {1, (1 + I)/Sqrt[2]}}

You could use Cases instead:

Cases[list, {1,Except[1]}|{Except[1],1}]

{{-1, 1}, {1, -1}, {1, (1 + I)/Sqrt[2]}}

Paranoid version of kglr's Count solution:

Select[list, Total[Boole[PossibleZeroQ[# - 1]] & /@ #] == 1 &]

{{-1, 1}, {1, -1}, {1, (1 + I)/Sqrt[2]}}