TITLE: Defining i,j,k,l,.. for complex number in a 3D space, in 4D space, etc
QUESTION [8 upvotes]: Real number is often used to represent a point in a 1-dimensional number line. Real numbers are written as $a$ where $a \in \mathbb{R}$.
Complex number is often used to represent a point in a 2-dimensional plane. Complex numbers are written as $a + bi$ where $a,b \in \mathbb{R}$ and $i^2 = -1$.
What about higher dimensional space?
We can extend the concept of complex number, and write a point in a 3-dimensional space as $a + bi + cj$ where $a,b,c \in \mathbb{R}$, but how do we define i and j?
Certainly, this also applies to 4-dimensional space. We can write a point in 4D as $a + bi + cj + dk$, how do we define $i,j,k$?
How about even higher n-dimensional space? Is there a system to define i,j,k,... that will extend to arbitrary dimensions? Is there a name for it?

REPLY [3 votes]: Over 20 years ago I found the way to generalize the complex numbers to higher dimensions.
Basically it is very simple: on $\mathbb{R}^3$ you define an imaginary unit $j$ via stating $j^3 = -1$.
Your complex numbers now look like $T = A + Bj + Cj^2$.

During the last 1.5 years I have picked up studying this stuff again, some of the results can be found at a page on my website:
3D Complex Stuff 
Compared to the complex plane the 3D complex numbers form a far more complicated structure; so not only university freshmen can learn from it, the same goes for retired math professors...
A nice starter exercise would be the next:
Use the above definition of 3D complex numbers (with $j^3 = -1$) and proof that the equation $X^2 = -1$ has no solution inside $\mathbb{R}^3$.