In a largely exploratory paper from 1987, "Self-reference and recursive forms" (J Social Biol. Struct. 1987, no. 10, 53-72), Louis H. Kauffman considers a number of different formal and semi-formal models for self-reference, but perhaps the most interesting aspect of the paper are the models in which he considers what would happen if you thought of truth values as working like complex numbers. In one of these (brief) discussions he uses the phrase in the title: "the square root of negation".
One of the models suggests that we could see the horizontal axis as representing 'simply true' (T, T) on the right and simply false (F, F) on the left. That is, we are considering two truth values -- for instance, perhaps our truth values represent the result we get on a test for whether something exists, and for checking purposes we do our test twice: (T, T) means that it passed on initial assessment and passed on check, (F, F) means that it failed on both. Then our vertical axis would be ambiguous cases: (T, F) if it passed on initial assessment and failed the check, which would be down, and (F, T) if it initially failed but passed the check, which would be up. As Kauffman suggests, you can see the imaginary axis as representing oscillating truth values, and the horizontal axis as consistent truth values. In this context, you need something like negation that tells you that we are not 'simply true' (T, T) but that does not carry you all the way across to 'simply false' (F, F), and yet that if you use it twice does do so. That is, you need something to serve the place of i, taking you a quarter turn rather than a half turn. We can, for instance, have an operation $ such that
$(A, B) = (-B, A)
$(T, T) = (F, T)
$(F, T) = (F, F)
$(F, F) = (T, F)
$(T, F) = (T, T)
In the Addendum to the paper he suggests that you could interpret (F, T) as a 'Possibly True' and (T, F) as 'Possibly False'. This wouldn't at all work, as is, on most uses of 'possibly', but Kauffman says, "As anyone intent upon the solution of a difficulty is actually aware, there is an enormous difference between attitudes of possible truth and possible falsehood" (p. 72). I'm not sure if that works, either, but I suspect Kauffman is thinking (as he does earlier) in terms of the coordination of multiple perspectives so really between (F, T) and (T, F) we are not keeping the same perspective but changing our perspective so it is looking at an ambiguous situation the other way around, i.e., from the other side of whatever ambiguity there might be. So instead of possibly we should perhaps say, "Ambiguously True" and "Ambiguously False". The primary issue with that is whether you want ambiguity to be symmetrical (that is, there is no real difference whichever way you look at it); perhaps something like Kauffman's argument seems relevant here: faced with an ambiguity, we can and do look at it from different directions, and it can matter. But perhaps there are reasons also to take the ambiguously false and the ambiguously true to be equivalent in general.
In any case, it's interesting to think of possible ways to make sense of the 'square root of negation'.