The expression for:
(x + i y)^{n}
where n is an integer.
is given by the following sum:
(x + i y)^{n}= 


(i)^{k} x^{nk} y^{k} 
We can derive this from the binomial theorm:
(a + b)^{n}= 


(1)^{k} a ^{nk} b^{k} 
with 'a' replaced by x and 'b' replaced by iy
Square
We can look at a simplest we can take the case where n=2 which gives:
(x + i y)^{2}
z plane  w plane  

> w=z² 
Pure real values always square to a positive value and pure imaginary values always square to a negative value. However real and imaginary parts together cover the whole plane.
Let the components of the input and output planes be:
z = x + i y and w = u + i v
lets take the example of the square function w = z²
so:
w = (x + i y)²
multiplying out gives:
w = x²  y² + i 2 x y
so the u and v components are:
u = x²  y²
v = 2 x y
Integer Powers
As shown above:
(x + i y)^{n}= 


(i)^{k} x^{nk} y^{k} 
where:
k  (i)^{k}  k! 

0  1  0 
1  i  1 
2  1  2 
3  i  6 
4  1  24 
So (i)^{k} cycles round every 4 entries, each step is a rotation by 90. The even values are real and the odd values are imaginary.
So the powers for n = 1 to 4 are:
n  (x + i y)^{n}  u^{}  v^{} 

1  (x + i y)^{1}  x  y 
2  (x + i y)^{2}  x²y²  2xy 
3  (x + i y)^{3}  x^{3}  3y²x  3x²y y^{3} 
4  (x + i y)^{4}  x^{4}  6x²y²  y^{4}  2y^{3}x2yx^{3} 