Higher dimensional equivalent of complex numbers If complex numbers are in some sense 2 dimensional numbers, would it be useful or logical to extend the complex number system to a system of 3-dimensional or even n-dimensional numbers? If this does not make sense in general, why does it make sense only to extend from the 1st dimension to the 2nd dimension? What is special about 2-dimensional numbers that they are needed but n-dimensional numbers are not? I know that we have $R^k$, which is in some sense a field of $k$ dimensional numbers, but that is very different from the way in which complex numbers are constructed. So my question is could the complex numbers be extended to a 3rd dimension (or even an nth dimension) in the same way that the real numbers are extended to the complex numbers, in which the properties of complex numbers are preserved as a subset of this higher dimensional space?
 A: It is not possible to define "higher-dimensional" analogues of $\mathbb{C}$ for every dimension $n$. 
It is not easy to show that though. One has a structure on $\mathbb{R}^4$ making it into a division ring - but it is not commutative anymore. 
Then there is for example a similar construction on $\mathbb{R}^8$ but the result will not be associative anymore.
A: It depends what you mean by "equivalent". You can invent an $n$-dimensional commutative extension of $\Bbb R$ viz. $\Bbb R[z]\setminus p(Z)$ with $\deg p=n$ ($\Bbb C$ is this with $p(Z)=Z^2+1$). But will you like the resulting number system? Division algebras need to be of dimension $1,\,2,\,4$ or $8$. You already know the first two, and the others don't commute (and the last one doesn't even associate), nor are they results of the above technique. There are also $2$-dimensional number systems that don't have inverses for all nonzero numbers (these two). Complex numbers are, in a way, the best balance of certain competing objectives - every advantage of $\Bbb R$ (except ordering), with algebraic completeness.
A: You need to specify which properties this new number system is required to possess; otherwise, you end up constructing something crazy and useless. 
Arguably (cfr. Steenrod), complex numbers should be called planar numbers, as their main feature is the representation of the planar similitudes as intuitive algebraic operations. The planar similitudes are the rotations, translations, changes of scale, and any combination of these three transformations. They constitute the symmetry group of the Euclidean plane geometry, and correspond to the algebraic operations of the complex numbers. 
Therefore, a good higher dimensional number system should be modeled on the group of rotations, translations and changes of scale in higher dimension. In dimension $3$, the quaternions are just that. Notice that, since three dimensional rotations are not commutative, the quaternionic number system is not commutative either. 
Things get worse in dimension higher than 3, though. I don't know any more details (see lush's answer for those).
A: Usually with even number of dimensions you have better results than with odd. Thus, you do not have anything useful or interesting in 3 dimensions. But in 4 dimensions you have multiple interesting algebras, some of which are more or less numbers-like.
Eevee Trainer has already mentioned quaternions, which are a division algebra but not commutative, but there are also other approaches which give algebras with different properties in 4D.

*

*First of all, possibly the simplest approach is to consider the $2\times2$ real matrices themselves, which is a 4D space. Matrix operations on $2\times2$ matrices are isomorphic to so-called split-quaternions:

$\begin{align}
\boldsymbol{1} =\begin{pmatrix}1&0\\0&1\end{pmatrix},\qquad&\boldsymbol{i} =\begin{pmatrix}0&1\\-1&0\end{pmatrix},\\
\boldsymbol{j} =\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad&\boldsymbol{k} =\begin{pmatrix}1&0\\0&-1\end{pmatrix}.\end{align}$
They are neither commutative, nor a division-algebra, but allow to apply any functions that can be generalized to square matrices. They have zero divisors, nilpotents and idempotents.


*Dual complex numbers. Formed by adding the dual unity $\varepsilon$ such that $\varepsilon^2=0$ to complex numbers. They can be represented by matrices of the form
$\left(
\begin{array}{cc}
 u & w \\
 0 & v \\
\end{array}
\right)$, where $u,v,w$ are complex numbers or matrices of the form $\left(
\begin{array}{cccc}
 a & b & c & d \\
 -b & a & -d & c \\
 0 & 0 & a & b \\
 0 & 0 & -b & a \\
\end{array}
\right)$ or $\left(
\begin{array}{cccc}
 a & c & b & d \\
 0 & a & 0 & b \\
 -b & -d & a & c \\
 0 & -b & 0 & a \\
\end{array}
\right)$ where $a,b,c,d$ are real numbers ($a$ is real part, $b$ is imaginary part, $c$ is dual part, $d$ is imaginary dual part). This system is commutative but has zero divisors and nilpotents.


*Tessarines. Formed by adding split-complex unity $j$ such as $j^2=1$ to complex numbers. Can be represented as matrices of the form $\left(
\begin{array}{cc}
 u & w \\
 w & u \\
\end{array}
\right)$, where $u,w$ are complex numbers or matrices of the form $\left(
\begin{array}{cccc}
 a & b & c & d \\
 -b & a & -d & c \\
 c & d & a & b \\
 -d & c & -b & a \\
\end{array}
\right)$ or $\left(
\begin{array}{cccc}
 a & c & b & d \\
 c & a & d & b \\
 -b & -d & a & c \\
 -d & -b & c & a \\
\end{array}
\right)$ where $a,b,c,d$ are real numbers ($a$ is real part, $b$ is imaginary part, $c$ is split-imaginary part, $d$ is the coefficient of $ij$ or $-ij=k$). This system is commutative algebra, but it has zero divisors and idempotents.
If one adds another complex unity $i_1$ such that $i_1^2=-1$ instead of split-complex unity $j$, one will get bicomplex numbers, which are isomorphic to this system (the split-complex unity will arise automatically as $j=-i i_1$).


*Dual split-complex numbers. They are formed by adding split-complex unity $j$ and dual unity $\varepsilon$ to real numbers. They can be represented as real matrices of the form $\left(
\begin{array}{cccc}
 a & b & c & d \\
 b & a & d & c \\
 0 & 0 & a & b \\
 0 & 0 & b & a \\
\end{array}
\right)$ or $\left(
\begin{array}{cccc}
 a & c & b & d \\
 0 & a & 0 & b \\
 b & d & a & c \\
 0 & b & 0 & a \\
\end{array}
\right)$ where $a,b,c,d$ are real numbers ($a$ is real part, $b$ is split-imaginary part, $c$ is dual part, $d$ is split-imaginary dual part).
This is a commutative algebra but with zero divisors, nilpotents and idempotents.


*4D split-complex numbers. These are formed by adding two split-complex unities $j$ and $j_1$ such that $j^2=j_1^2=1$. They are isomorphic to the matrices of the form $\left(
\begin{array}{cccc}
 a & b & c & d \\
 b & a & d & c \\
 c & d & a & b \\
 d & c & b & a \\
\end{array}
\right)$ or $\left(
\begin{array}{cccc}
 a & c & b & d \\
 c & a & d & b \\
 b & d & a & c \\
 d & b & c & a \\
\end{array}
\right).$ This is a commutative system, with zero divisors and idempotents.


*4D dual numbers. These are formed by adding two dual unities to real numbers: $\varepsilon$ and $\varepsilon_1$, such that $\varepsilon^2=\varepsilon_1^2=\varepsilon\varepsilon_1=0$. They are isomorphic to the real matrices of the following form:
$\left(
\begin{array}{cccc}
 a & b & c & d \\
 0 & a & 0 & c \\
 0 & 0 & a & b \\
 0 & 0 & 0 & a \\
\end{array}
\right)$. This system is commutative but has zero divisors and nilpotents.
