Imaginary numbers and Quantum mechanics

[Arp220]

Ok, so this has always puzzled me a bit. Tell me what you think.

Consider the following:

Over a century ago, some geeky mathematician types were sitting around, and idly thinking about number theory. Or something. And they got to chatting about square roots and cube roots and so on. And realised that mathematics as they knew it had no way to deal with the root of a negative number. So they decided address this by calling the square root of '1 'i'. And then had a grand old time messing about with this utterly arbitrary, completely theoretical mathematical construction that had no practical application and no basis in reality whatsoever.

And then, a few decades later, a whole bunch of scientific thingies came along, most notably quantum mechanics, but also things like signal processing, where imaginary numbers were essential to their success.

Now, we can all see examples where whole numbers, fractions, percentages, negative numbers, etc etc are 'real' - for example credits and debts require negative numbers, computing ares requires fractions (usually), and so on.

But then a whole bunch of sciency stuff comes along that absolutely requires imaginary numbers for their theoretical underpinnings, despite the fact that imaginary numbers were devised as little more than an intellectual masturbation exercise several decades earlier, with no though that they represented, or indeed could represent, anything 'real'.

So what puzzles me is this - by this logic, imaginary numbers are also real. But in what sense are they real? What aspect of reality does the square root of minus one describe?