A complex number \(z\) can be written as
$$ z = a + i b; $$
where both \(a\) and \(b\) are real numbers, and \(i^{2} = -1\). Another way to represent a complex number is as a pair of real numbers:
$$ z = \begin{pmatrix} a & b \end{pmatrix}. $$
I will prefer this way since it is more compatible with how a complex number is represented computationally. The left component is called the real part:
$$ \operatorname{Real}{(z)} = a; $$
and the right component is called the imaginary part:
$$ \operatorname{Imag}{(z)} = b. $$
Instead of "imaginary" I will use the term "unreal" for the part of a complex number that is not real.
Addition and Scaling
You can define two operations on complex numbers. The addition operation is given by
$$ z_{1} = \begin{pmatrix} a_{1} & b_{1} \end{pmatrix}, \qquad z_{2} = \begin{pmatrix} a_{2} & b_{2} \end{pmatrix} \quad \longrightarrow \quad \operatorname{Add}{(z_{1}, z_{2})} = \begin{pmatrix} a_{1} + a_{2} & b_{1} + b_{2} \end{pmatrix}. $$
The scale operation is given by
$$ z = \begin{pmatrix} a & b \end{pmatrix} \quad \longrightarrow \quad \operatorname{Scale}{(z, c)} = \begin{pmatrix} c a & c b \end{pmatrix}. $$
The addition and scale operations are essential for vector spaces. Indeed, you can assign a complex number to a 2-dimensional real vector and vice-versa:
$$ \mathbf{v} = \begin{bmatrix} a & b \end{bmatrix} \quad \longleftrightarrow \quad z = \begin{pmatrix} a & b \end{pmatrix}. $$
Conjugation and Negation
You can have two operations on complex numbers that translate to discrete transformations of 2-vectors. The first is the conjugation operation:
$$ z = \begin{pmatrix} a & b \end{pmatrix} \quad \longrightarrow \quad \operatorname{Conj}{(z)} = \begin{pmatrix} a & -b \end{pmatrix}. $$
The second is the negation operation:
$$ z = \begin{pmatrix} a & b \end{pmatrix} \quad \longrightarrow \quad \operatorname{Neg}{(z)} = \begin{pmatrix} -a & -b \end{pmatrix}. $$
Both conjugation and negation are involutions:
$$ \operatorname{Conj}{(\operatorname{Conj}{(z)})} = z; \qquad \operatorname{Neg}{(\operatorname{Neg}{(z)})} = z. $$
Subtraction
With negation and addition you can define the subtraction operation:
$$ z_{1} = \begin{pmatrix} a_{1} & b_{1} \end{pmatrix}, \qquad z_{2} = \begin{pmatrix} a_{2} & b_{2} \end{pmatrix} \quad \longrightarrow \quad \operatorname{Sub}{(z_{1}, z_{2})} = \operatorname{Add}{(z_{1}, \operatorname{Neg}{(z_{2})})} = \begin{pmatrix} a_{1} - a_{2} & b_{1} - b_{2} \end{pmatrix}. $$
Multiplication
There are two ways to multiply two 2-dimensional real vectors and obtain a single number. You have the dot product:
$$ \mathbf{v} = \begin{bmatrix} v_{1} & v_{2} \end{bmatrix}, \qquad \mathbf{w} = \begin{bmatrix} w_{1} & w_{2} \end{bmatrix} \quad \longrightarrow \quad \operatorname{Dot}{(\mathbf{v}, \mathbf{w})} = v_{1} w_{1} + v_{2} w_{2}; $$
and the cross product:
$$ \mathbf{v} = \begin{bmatrix} v_{1} & v_{2} \end{bmatrix}, \qquad \mathbf{w} = \begin{bmatrix} w_{1} & w_{2} \end{bmatrix} \quad \longrightarrow \quad \operatorname{Cross}{(\mathbf{v}, \mathbf{w})} = v_{1} w_{2} - v_{2} w_{1}. $$
Note that
$$ \operatorname{Dot}{(\mathbf{v}, \mathbf{w})} = \operatorname{Dot}{(\mathbf{w}, \mathbf{v})}, \qquad \operatorname{Cross}{(\mathbf{v}, \mathbf{w})} = -\operatorname{Cross}{(\mathbf{w}, \mathbf{v})}. $$
There is only one way to multiply two complex numbers and get another complex number:
$$ z_{1} = \begin{pmatrix} a & b \end{pmatrix}, \qquad z_{2} = \begin{pmatrix} c & d \end{pmatrix} \quad \longrightarrow \quad \operatorname{Mul}{(z_{1}, z_{2})} = \begin{pmatrix} ac - db & da + bc \end{pmatrix}. $$
This is the multiplication operation. Note that
$$ \operatorname{Mul}{(z_{1}, z_{2})} = \operatorname{Mul}{(z_{2}, z_{1})}. $$
That is, the multiplication operation for complex numbers is commutative.
The 2-dimensional dot and cross products can be recovered from
$$ \operatorname{Mul}{(\operatorname{Conj}{(z_{1})}, z_{2})} = \begin{pmatrix} ac + db & da - bc \end{pmatrix}. $$
If you have two real 2-vectors
$$ \mathbf{v} = \begin{bmatrix} v_{1} & v_{2} \end{bmatrix}, \qquad \mathbf{w} = \begin{bmatrix} w_{1} & w_{2} \end{bmatrix}; $$
and two complex numbers
$$ z_{1} = \begin{pmatrix} v_{1} & v_{2} \end{pmatrix}, \qquad z_{2} = \begin{pmatrix} w_{1} & w_{2} \end{pmatrix}; $$
then
$$ \operatorname{Dot}{(\mathbf{v}, \mathbf{w})} = \operatorname{Real}{(\operatorname{Mul}{(\operatorname{Conj}{(z_{1})}, z_{2})})}, \qquad \operatorname{Cross}{(\mathbf{v}, \mathbf{w})} = \operatorname{Imag}{(\operatorname{Mul}{(\operatorname{Conj}{(z_{1})}, z_{2})})}. $$
Lozenge
With the multiplication and conjugation operations, you can define the lozenge operation:
$$ \operatorname{Lozenge}{(w,x,y,z)} = \operatorname{Mul}{(w, \operatorname{Conj}{(x)})} - \operatorname{Mul}{(y, \operatorname{Conj}{(z)})}. $$
This operation is useful when working with quaternions as pairs of complex numbers.
Quadrance
The square magnitude of a real 2-vector is given by
$$ \mathbf{v} = \begin{bmatrix} v_{1} & v_{2} \end{bmatrix} \quad \Longrightarrow \quad \operatorname{Quad}{(\mathbf{v})} = v_{1}^{2} + v_{2}^{2}. $$
This is also known as the elliptic quadrance. The elliptic quadrance is always non-negative. A complex number also has a quadrance:
$$ z = \begin{pmatrix} a & b \end{pmatrix} \quad \Longrightarrow \quad \operatorname{Quad}{(z)} = \operatorname{Mul}{(\operatorname{Conj}{(z)}, z)} = a^{2} + b^{2}. $$
The quadrance of a complex number is also always non-negative.
Inversion and Division
Since the quadrance of a complex number is a non-negative real number, in general you can define the inversion operation by
$$ \operatorname{Inv}{(z)} = \operatorname{Scale}{(\operatorname{Conj}{(z)}, 1/ \operatorname{Quad}{(z)})}. $$
The only time when this fails is when the quadrance is zero. The only complex number that has zero quadrance is the zero complex number:
$$ z = \begin{pmatrix} 0 & 0 \end{pmatrix}. $$
Division can be defined in terms of multiplication and inversion. The division operation is
$$ \operatorname{Quo}{(z_{1}, z_{2})} = \operatorname{Mul}{(z_{1}, \operatorname{Inv}{(z_{2})})}. $$
Since the multiplication operation is commutative, the division operation is unique:
$$\operatorname{Mul}{(z_{1}, \operatorname{Inv}{(z_{2})})} = \operatorname{Mul}{(\operatorname{Inv}{(z_{2})}, z_{1})}$$
That is, there is no difference between left or right quotients.
The division operation can be define in a way that is analogous to the multiplication operation:
$$ z_{1} = \begin{pmatrix} a & b \end{pmatrix}, \qquad z_{2} = \begin{pmatrix} c & d \end{pmatrix} \quad \longrightarrow \quad \operatorname{Quo}{(z_{1}, z_{2})} = \begin{pmatrix} \dfrac{ac + db}{c^{2} + d^{2}} & \dfrac{bc - da}{c^{2} + d^{2}} \end{pmatrix}. $$
Möbius Transformation
Given a complex number \(z\) and four other complex numbers \(a\), \(b\), \(c\), and \(d\), the Möbius transformation of \(z\) is given by
$$ \operatorname{Mobius}{(z, a,b,c,d)} = \frac{a z + b}{c z + d}. $$
Cross-Ratio
Given four complex numbers \(w\), \(x\), \(y\), and \(z\), the cross-ratio is given by
$$ \operatorname{CrossRatio}{(w,x,y,z)} = \frac{(w - y)(x - z)}{(x - y)(w - z)}. $$
The cross-ratio is invariant under Möbius transformations.
Differentiation
You can use two real variables \(x\) and \(y\) to construct a complex variable \(z\):
$$ z = x + i y = \begin{pmatrix} x & y \end{pmatrix}. $$
Similarly, you can use two real differential operators \(\partial_{x}\) and \(\partial_{y}\) to construct a complex differential operator:
$$ \partial \equiv \frac{1}{2} \left( \partial_{x} - i \partial_{y} \right). $$
This operator is sometimes called the Wirtinger derivative. The Wirtinger derivative satisfies
$$ \partial z = 1, \qquad \partial \operatorname{Conj}{(z)} = 0, \qquad \operatorname{Conj}{(\partial)} z = 0, \qquad \operatorname{Conj}{(\partial)} \operatorname{Conj}{(z)} = 1. $$
A more useful operator is the complex gradient operator:
$$ \nabla \equiv \partial_{x} + i \partial_{y}. $$
In terms of real 2-vectors, you have the position 2-vector:
$$ \mathbf{r} = \begin{bmatrix} x & y \end{bmatrix}. $$
and the gradient 2-vector:
$$ \nabla = \begin{bmatrix} \partial_{x} & \partial_{y} \end{bmatrix}. $$
A function 2-vector has two real functions as components:
$$ \mathbf{F}(\mathbf{r}) = \begin{bmatrix} F_{x}(x, y) & F_{y}(x, y) \end{bmatrix}. $$
Two important spatial derivatives are the euclidean divergence:
$$ \nabla \cdot \mathbf{F} = \partial_{x} F_{x} + \partial_{y} F_{y}; $$
and the euclidean curl:
$$ \nabla \times \mathbf{F} = \partial_{x} F_{y} - \partial_{y} F_{x}. $$
Note that for real euclidean 2-vectors, you have
$$ \nabla \cdot \mathbf{r} = 1 + 1 = 2; \qquad \nabla \times \mathbf{r} = 0. $$
Writing the function 2-vector as a complex number,
$$ F = \begin{pmatrix} F_{x}(x, y) & F_{y}(x, y) \end{pmatrix}; $$
you have
$$ 2 \operatorname{Real}{(\partial F)} = \operatorname{Real}{(\operatorname{Conj}{(\nabla)} F)} = \nabla \cdot \mathbf{F}, \qquad 2 \operatorname{Unreal}{(\partial F)} = \operatorname{Unreal}{(\operatorname{Conj}{(\nabla)} F)} = \nabla \times \mathbf{F}. $$
Assuming that the components of \(F\) are smooth, you also have
$$ 4 \operatorname{Conj}{(\partial)} \partial F = \operatorname{Conj}{(\nabla)} \nabla F = \begin{pmatrix} \partial_{x}^{2} F_{x} + \partial_{y}^{2} F_{x} & \partial_{x}^{2} F_{y} + \partial_{y}^{2} F_{y} \end{pmatrix}. $$
