What is lost when we move from reals to complex numbers?

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.

The most important property you loose when moving from real to complex numbers is definitely the notion of an order, i.e. $\mathbb{R}$ is an ordered field whereas $\mathbb{C}$ is not. This follows from the following proposition (Abstract Algebra by P.A. Grillet):

A field $F$ can be ordered if and only if $-1$ is not a sum of squares of elements of $F$.

Moreover, we have to be careful in defining certain functions due to the fact that now even standard functions turn out to be multi-valued rather than single-valued. However, this is paid back immediately by the nice differentiability properties of complex differentiable functions.

Losing order is the most important but we also lose that if $b > 1$ then $b^z$ is injective so $\ln z$ (or $\log_b z$) is no longer a function but an equivalence class.