Extending the set of complex numbers

To solve polynomials, the answer is no. The Fundamental Theorem of Algebra says that every (non-constant) polynomial, of degree $n$, with complex coefficients has $n$ roots (including repeated roots) in $\mathbb{C}$.

As a result of this, the set of complex numbers, unlike the set of real numbers, is algebraically closed, which means that we cannot 'escape' $\mathbb{C}$ using any elementary operations, like $+, - , \times, \div, \sqrt{}, e^{...}$ etc. So, in this sense, we don't really 'need' to extend the complex numbers.

However, there do exist hypercomplex numbers, like the quaternions, $\mathbb{H}$, which, instead of using just one imaginary unit $i$, use three: $i, j, k$, each satisfying:$$i^2=j^2=k^2=ijk=-1$$

Quaternions are used in modelling 3D vectors, and have a lot of use in 3D mechanics. A useful property of a quaternion is that it does not satisfy commutativity: e.g. $i\times j \neq j \times i$.

N0w, if you still want to go further, you can explore the octonions, $\mathbb{O},$ which are an extension of the quaternions. Octonions have 7 imaginary units: $e_1, e_2, ...e_7$.

Octonions have far fewer (that I know of, at least) practical uses that quaternions. Octonions also have the interesting property that they lack associativity: e.g. $x\times(y\times z) \neq (x \times y) \times z$ (for $x,y,z \in \mathbb{O}$).

And, finally, at least as far as mathematical interest has gone, we have the sedenions, $\mathbb{S}$. These are an extension of the octonions and have 15 imaginary units: $e_1,e_2,e_3,...,e_{15}$. The special property about the sedenions is that they have zero-divisors (meaning that there exist non-zero sedenions $x$ and $y$ such that $xy=0$). Interesting, this property, isn't it? Not particularly, intuitive, to say the least.

Now, I'll leave as an exercise for you to find out about the applications of hypercomplex numbers, and how to multiply them (hint: Google 'Fano plane mnemonic', which explains how to multiply octonions- this same idea can be extended to the sedenions).

See http://en.wikipedia.org/wiki/Hypercomplex_number

As an account of history the question is nonsense. Real numbers and complex numbers were thought about before Cantor's birth (he was born in the 19th century), and solving algebraic equations with no real solutions was not the motive.

Complex numbers were introduced in the 16th century in order to find real solutions of third-degree algebraic equations with real coefficients.

That sort of process for solving algebraic equations will not take you beyond the complex numbers, because a theorem, often misleadingly called the "fundamental theorem of algebra" says that every algebraic equation with complex coefficients has a complex solution.

But, as others have already noted here, there are other motives for extending the system of complex numbers. The quaternions are the most widely known example.

You can extend the complex numbers and get quaternions. Which can be used in, for example, 3D computer graphics instead of matrices. It is possible to extend the quaternions as many times as you want, like you did with the complex numbers to get the quaternions.