This isn't a great comparison, the problem with dividing by zero is it could be literally anything, it is ambiguous. To have zero in the denominator is to say "I haven't provided you the necessary information to extract a value from this ratio, therefore the value could be anything." Like if you were to say 1/2. I am saying "I have divided something exactly once evenly to produce two equal parts, and I have handed you 1 of those parts." 0/2 is saying "I divided something exactly once evenly to create two equal parts and I have given you zero of them." 1/0 is saying "Something was divided into equal parts but I don't know how many times and here is one of them." <-- that isn't enough information to create a rational number.

The square root of negative 1 is saying "in a real sense negative square roots can't exist, but if they did exist we can manipulate them in this way." It is similar to anything in a computer that is 'virtual', it literally doesn't exist but if we imagine they do we can still do interesting thing.