Comment by OldRonin

The field C is a tool, just as with any other abstraction in math. Accountants don't need it, so ignore it. In abstract algebra they arise - and so do many other abstractions, such as quaternions. Quaternions can be regarded as a further development of the concept of number. Note that every such development gives up some property its predecessors had. In the case of quaternions, commutativity. If a field of application finds a mathematical invention useful, it gets used. Often, too, it's a matter of simplicity. Use mathematical invention X in applied field Y and it simplifies the work. Abstain from using invention X, and then the work could still be done in a less simple way. That's all there is to it.