What is lost when we move from reals to complex numbers? As I know when you move to "bigger" number systems (such as from complex to quaternions) you lose some properties (e.g. moving from complex to quaternions requires loss of commutativity), but does it hold when you move for example from naturals to integers or from reals to complex and what properties do you lose?
The most important ones as I see it:


Naturals to integers: lose well-orderedness, gain "abelian group" (and, indeed, "ring").
Integers to rationals: lose discreteness, gain "field".
Rationals to reals: lose countability, gain "Cauchy-complete".
Reals to complexes: lose a compatible total order, gain the Fundamental Theorem of Algebra.
$\mathbb C$ is more rigid. Holomorphic functions are analytics. This means complex manifolds are polynomial-like and more akin to algebraic varieties than to real manifolds. By contrast $\mathbb R$ manifolds can be glued together using functions like $\exp(-1/x^2)$ that can $C^\infty$-smoothly transition from one function to another.