In mathematics, polynomials like $x^2-1$ would have a clear solution of $x=\pm 1$. However, without complex numbers could you solve $x^2+1$? No, there would be no possible solution without adding a new axis of imaginary numbers. Simply, we added this axis because it provided us with solutions within our field of mathematics, so then we assumed them to exist.
So we know that you can only assume something to exist if it produces a solution (the most important reason for the imaginary axis to exist), then my question is why don't we do the same for an additional axis called $j$?
For example, we created the imaginary numbers by $\sqrt{-1}=i$, which then we unioned this imaginary axis to our real number axis to form the complex plane $\mathbb{C}=a+bi$. So then, why can't we square root a negative complex number to equate $j$. Visually:
\begin{gather*}
\sqrt{-(a+bi)}=j\\
\textrm{We then union this new axis with the previous to form the new set of numbers:}\\
\mathbb{C}_2=a+bi+cj:a,b,c\in\mathbb{R}\\\\
\textrm{We can then further extend this to even greater numbers:}\\
\mathbb{C}_3=a+bi+cj+dk: a,b,c,d\in\mathbb{R}\\
\mathbb{C}_\infty=a+bi+cj+dk+…+z(\infty): a,b,c,d,…,z\in\mathbb{R}
\end{gather*}
And this repeats on and on, where you take the square root of a negative number and from this method you keep on formulating a new type of axis of a number.
Why don't we already have this in mathematics? We've done it once with solely the imaginary plane, why not extend it to a three axis field? Why do we limit ourselves?
I'm more than sure there are problems out there that can be solved with a $\mathbb{C}_2$ number!
I must say the beauty of this is mathematically speaking we can then equate infinity or at least have a new symbol for it. $\mathbb{C}_\infty=\infty$
Best Answer
To answer your question: the complex numbers are algebraically closed. But this should not stop you from defining new "complex" numbers, such as quaternions, octonions, sedenions etc. This is called the Cayley-Dickson construction.