Defining i,j,k,l,.. for complex number in a 3D space, in 4D space, etc 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?
It depends upon what structure you want.  In $\mathbb{R}^n$ we do not have the operation of multiplying two elements, as we do on $\mathbb{C}$. If you want that, you probably want a normed division algebra, which are available in four and eight dimensions, but no more.

If you just want a vector space they are basis vectors but you can't multiply two elements.
If we take complex number in 3d plan than we found 2 condition,
let we take 3 axis x y and z:


x is real and yand z are imaginary, that is: $x+i(y+z)$
x and z is real but y is imaginary, that is: $x+z+iy$