Square roots of negative identity

sqrt(-I) for I=1 is not defined in the reals. However, for I=eye(2), there are solutions. For example [0 1 ; -1 0]² = [-1 0 ; 0 -1]. The general solution in reals (thanks to Matlab) seems to be [-d, b ; -(1+d²)/b, d]. For 3x3 matrices there appear to be no solutions, so it seems like an odd-even dimensions thing.