What do we lose passing from the reals to the complex numbers?

Question

As normed division algebras, when we go from the complex numbers to the quaternions, we lose commutativity. Moving on to the octonions, we lose associativity. Is there some analogous property that we lose moving from the reals to the complex numbers?

Answer 1

The most obvious property that we lose is the linear (or total) ordering of the real line.

Answer 2

We lose equality of the complex conjugate and total order.

So: $x+i y \ne x-i y$ for complex numbers which are not also reals.

And you can't say wether $ i > -i $ or $i < -i$, etc.
All you have is the magnitude which, in the given example, is equal.