distance measures

Which of the cosmological distance measures are additive, \(D(z_1,z_2) + D(z_2,z_3) = D(z_1,z_3)\) and which ones are monotonic, \(dD/dz > 0\)?

bonus points, like always: What’s \(\mathrm{i}^\mathrm{i}\) with \(\mathrm{i} = \sqrt{-1}\)?