I created a program in python that generates an image of the mandelbrot set. The only problem I have is that the program is quite slow, it takes about a quarter of an hour to generate the following image of 2000 by 3000 pixels:

I first created a matrix of complex numbers using numpy according to amount of pixels. I also created an array for the image generation.

import numpy as np
from PIL import Image

z = 0

real_axis = np.linspace(-2,1,num=3000)
imaginary_axis = np.linspace(1,-1,num=2000)

complex_grid = [[complex(np.float64(a),np.float64(b)) for a in real_axis] for b in imaginary_axis]

pixel_grid = np.zeros((2000,3000,3),dtype=np.uint8)

Then I check whether each complex number is in the mandelbrot set or not and give it an RGB colour code accordingly.

for complex_list in complex_grid:
    for complex_number in complex_list:
        for iteration in range(255):
            z = z**2 + complex_number
            if (z.real**2+z.imag**2)**0.5 > 2:
                pixel_grid[complex_grid.index(complex_list),complex_list.index(complex_number)]=[iteration,iteration,iteration]
                break
            else:
                continue
        z = 0

Finally I generate the image using the PIL library.

mandelbrot = Image.fromarray(pixel_grid)
mandelbrot.save("mandelbrot.png")

I am using jupyter notebook and python 3. Hopefully some of you can help me improve the performance of this program or other aspects of it.