The Maths
The letter i represents the square root of -1 √-1. It is imaginary. You cannot multiply two identical numbers, even negatives and get a negative answer.
However, i2 = -1 or √-1 x √-1 = -1
Using this 'Other World' number is the base of the fantastic graphics of Mandelbrot
A computer can calculate thousands of coordinates in a second. We'll choose a few examples of FIRST coordinates and their MOVE stability. The maths formula is the same for all coordinates.
Coordinates are written (X , Yi). Y values are multiplied by i. Each FIRST coordinate is converted to a complex number X+Yi and squared (X+Yi)2 . We square X plus Yi by ordinary algebra
(X+Yi)2 = X2 + (Yi)2+2(XYi)
Starting with an example FIRST coordinate and its complex number 0.2 + 0.4i. Squaring this we get...
(0.2+0.4i)2 = 0.22+(0.4i)2+2(0.2 x 0.4i)
The Yi2 term always includes i2 (real number, -1) and goes from our pink zone to the real.
Our square result is a complex number
-0.12 + 0.16i
The FIRST complex number 0.2 + 0.4i is added to the square result -0.12 + 0.16i giving 0.08 + 0.56i and its MOVE coordinate (0.08 , 0.56i).
The FIRST complex number, in this case 0.2 + 0.4i , will be added to each subsequent MOVE squared.
Second move: On your own, take this first MOVE coordinate (0.08 , 0.56i), square it as a complex number (0.08+0.56i)2, to get a square result. Add this to the FIRST complex number 0.2 + 0.4i.