Are there any numbers more fundamental than Complex numbers? Are there any numbers other than Complex Numbers which contain something more that Real and Imaginary numbers?
 A: The Quaternions $\mathbb H$ maybe? They are non-commutative, though.
A: The Cayley Dickson construction might be a useful related remark. 
A: Yes.  There is in fact a well-structured set of numbers of dimension $2^n$ for all $n$, but after a certain point it simply becomes pointless and unwieldy.  The next set of numbers is called the quaternions of the form $a+bi+cj+dk$, where $a,b,c,d\in\mathbb{R}$ and $i,j,k$ are all square roots of $-1$.  In the quaternions, $ij=k, jk=i, ki=j$ but $ji=-k, kj=-i, ik=-j$ (that's right, they're non-commutative).  The complex numbers $\mathbb{C}$, though, are considered more fundamental than the quaternions $\mathbb{H}$ or the sets that follow (the octonions, sedenions, etc.), since the complex numbers are an abelian field with algebraic closure (algebraic closure of a field $F$ means that all polynomials $\sum a_ix^i$ with $a_i\in F$ have all of their roots in $F$.  This cannot be said of $\mathbb{R}$, since $x^2+1$ has real coefficients but imaginary roots.), giving it the greatest structure of all of these number sets.  Funny things start to happen after the quaternions, as the octonions are no longer associative and the sedenions can have zero divisors (which means two non-zero elements $a$ and $b$ can have the property $ab=0$).
A: Going off in a slightly different direction than sets containing the complex numbers are the hyperreals ${}^\ast\mathbb{R}$, which is used in non-standard analysis.  It can be thought of all adding infinitesimals (numbers that are unequal to zero but smaller in magnitude then any non-zero real number) and infinitely large numbers (larger in magnitude than any number).  The are the rigorous formulation of the intuitions using infinitesimals and indivisibles before the $\epsilon-\delta$-definitions were used to bypass the limited rigor employed at the time.  What's nice about this is that it remains a field, and is in a sense equivalent to the reals through the "Transfer principle", which relates first-order sentences in ${}^\ast\mathbb{R}$ to those in $\mathbb{R}$.
