Hmm.. Imaginary numbers are indeed a bit confusing.

I'm trying to imagine a 2d surface where the X-axis coordinates are all the real numbers, and the y axis are all the imaginary numbers. That makes them orthogonal, and that seemed to add up with your explanation, up until ixi=-1.

The only way I can get that to add up is if I instead imagine a arbitrary coordinate system, where x and y are not necessarily perpendicular, and i describes the angle between x and y.

I've only just finished my first cup of coffee for the day, so I haven't quite decided yet if that makes any sense whatsoever, but it's the only way I can intuit about it that includes a circular motion like the one you describe..

In this case you could almost describe i as the square root of 180°, which... Yeah it's kinda weird...

Am I still on the right track?