Multidimensional complex systems with 3, 4 or more dimensions are constructed. They possess algebraic operations which have geometrical meanings. Multidimensional complex numbers can be written in Cartesian, trigonometric and exponential form and can be converted from one form to another. Each complex numbers has a conjugate. Multidimensional complex systems are extensions of the classical complex number system.

In about 500 years after the birth of complex number, there were several attempts to extend complex number to more than 2 dimensions, for example we have theories such as quaternions, tessarines, coquaternions, biquaternions, and octonions. But none has reached the success of the classical complex number in 2 dimensions. Among these theories the most famous is quaternion which has found use in computational geometry. But quaternion is a 4 dimensional complex number but is used in 3 dimensional vector space, which is somewhat awkward.

In this article we will show that multidimensional complex number with 3, 4 or more dimensions exist and will explain how to construct them. Like classical complex number system, a multidimensional complex number system possesses algebraic operations in its complex space that have geometrical meaning in the corresponding vector space.

In the following exposition, spaces with 3, 4 or n dimensions will be referred to as 3D, 4D and nD spaces and the corresponding complex numbers as 3D, 4D and nD complex numbers. Since a complex number corresponds to a vector, a complex number will be referred to as a vector when convenient. We will begin with constructing 3D complex number system. Then we will generalize to spaces with 4 and more dimensions.

The 3D complex number system is constructed from a 2D complex number system which is the classical complex number system. So, let us see how classical complex number works.

Classical complex number

Classical complex space is a plane with two orthogonal axes, see Figure 1:

The axis of real numbers which is labeled as h.

The axis of imaginary numbers which is labeled as i.

This plane is labeled as (h, i). On this plane a complex number is both a point and a vector, for example the vector u in Figure 1. u makes the angle with the axis h and and its length is |u|. As complex number, u’s argument is and its modulus is |u|. In polar coordinate system the complex number u is expressed in equation (1), where i is the imaginary unit, see (3). Equation (1) is referred to as the trigonometric form of u.

We develop (1) into (2) in which we introduce (4) and obtain (5) where the numbers ‘a’ and b are the Cartesian coordinates of u. So, equation (5) is referred to as the Cartesian form of u.

Equation (6) is the Euler’s formula for and i, and is introduced into (1) which becomes (7). Equation (7) expresses u in the form of an exponential function and is referred to as the exponential form of u. So, a classical complex number can be expressed in Cartesian, trigonometric or exponential form and has a geometrical meaning which is the vector u in Figure 1.

3D complex number

3D space and vector

A 3D complex number is also a vector, which we will construct from the 2D plane (h, i). For doing so, we add the axis j perpendicularly to the plane (h, i) and obtain the 3D space whose axes are labeled as h, i and j, see Figure 2. We refer to this space as (h, i, j).

We attach the unit vectors eh, ei and ej to the axes h, i and j respectively. The 3D space based on these vectors is referred to as (eh, ei, ej). We have then two 3D spaces: the complex space (h, i, j) and the vector space (eh, ei, ej). We will create a vector labeled as v in (eh, ei, ej) which corresponds to a 3D complex number in (h, i, j) labeled also as v.

With the help of Figure 2 we create the vector v in the desired form by starting with a vector u which is expressed in (8) with |u| being its modulus and the angle it makes with the axis h. So, u is in the horizontal plane (eh, ei). Dividing u by |u| gives the unit vector eu, see (9). The unit vectors eu and ej are the basis vectors of the vertical plane (eu, ej), see Figure 2. The vector v is created by rotating the vector u in this plane toward the axis j. The angle of rotation is , so v is expressed with the angle on the basis vectors eu and ej in (10).

As the length of u stays the same during the rotation, the modulus of u and v are equal, see (11). Introducing the expression of eu (9) into (10) gives (12) which is developed into (13) using (11). The vector v is expressed with its modulus |v| and the angles and on the basis vectors eh, ei and ej, see (13).

Notice that the angle is between the vector v and the horizontal plane (eh, ei), see Figure 2, which is different from the usual spherical coordinate system where the angle is between the vector v and the axis j. So, when u is horizontal the angle equals zero rather then /2.

See the article with figures and equations
here

https://www.academia.edu/71708344/Extending_complex_number_to_spaces_with_3_4_or_any_number_of_dimensions
https://pengkuanonmaths.blogspot.com/2022/02/extending-complex-number-to-spaces-with.html