When are complex numbers insufficient?

Question

Rationals can't solve $x^2=2$, and reals can't solve $x^2=-1$. Is there any problem that cannot be solved by complex numbers but can be solved by non-standard numbers?

Every polynomial with coefficients in $C$ can be solved by numbers in $C$, can every equation* be solved by numbers in $C$?

*(that can not be simplified to $1=0$)

Answer 1

As $\mathbb{C}$ is commutative, non-commutative problems cannot be solved in it.

E.g. $A \cdot B - B \cdot A = I.$

Equations like this are central to Quantum Mechanics and Lie Algebras.

However, matrices usually are non-commutative, so matrices over $\mathbb{C}$ or $\mathbb{R}$ are able to solve those equations. As can quaternions among others.

EDIT: You say that you are looking for non-standard numbers. May I suggest to have a look at Quaternions? They are the logical next step after $\mathbb{C}$. The Wikipedia entry might be a good starting point. Quaternions are a bit out of fashion, but theoretically and historically they are important.

Answer 2

The problem, "find a number strictly greater than zero but strictly less than every number $1/n$ with $n=1,2,3,\dots$" cannot be solved in the complex numbers, but can be solved in the non-standard numbers.