Thank you hanciong and Merrak. I'll digest all that at home. It's just a case of learning a different set of rules.

BTW, The riddle solution goes up on Friday, 5th July. It's not too late to submit an answer.

Would a good metaphor be the z plane in 3D software which doesn't actually exist on a 2D screen but helps us to visualise real structures?

GC is back!

Hmmm... I'm not sure how good of a metaphor that is. For one thing, the z plane still "exists" - it just isn't seen in a top-down projection. I'm having trouble thinking of a good metaphor.

Maybe an airplane in flight would be good. A car on a trip is only concerned with its distance from the starting point. A plane, however, would be concerned with its distance from the start and its height.

This actually gets back to the grid I had in my blog post. A complex number, like 3 + 5i, is represented by a point some distance left/right and some distance up/down from the starting point.

As another example of changing rules, a seemingly simple concept such as "greater than" suddenly becomes complicated. Pick the bigger number: 3 + 5i or 2 + 37i.

There are two common ways to think about it, and they both give a different result

If you use the graph I had in my blog post, and go with distance from the origin, 2 + 37i is bigger. But another common approach is to use "dictionary order". Just like "Bobcat" comes after "Applesauce" in the dictionary, 3 + 5i would come after 2 + 37i - since 3 > 2. For complex numbers, distance from zero is the correct approach. But the alternative (which is a different number system) does have its uses.