TITLE: Extending the set of complex numbers
QUESTION [11 upvotes]: Mathematics as a science became richer when Cantor invented the real numbers. Then scientists wanted to solve equations which were not solvable in the real numbers so they invented the complex numbers. My question is, will we ever need to extend the complex numbers to some other set which will contain all other existing sets and help us for something? Are complex numbers extendable? Thanks in advance.

REPLY [3 votes]: Others have already mentioned the quaternions.  I just wanted to mention the surcomplex numbers, which, while lacking the property of being a set, satisfy all the properties of an algebraicly closed field.  You get these numbers by taking the surreal numbers (a gigantic totally ordered field containing every totally ordered field in existence) and adjoining the element $i$, which is analagous to how you construct $\mathbb{C}$ by adjoining $i$ to $\mathbb{R}$.