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This is a minor change, with a 50% improvement to @gareth's answer. Changing

escaped = abs(z_live) > threshold to

escaped = z_live.real**2 + z_live.imag**2>threshold**2

while less nice looking is about 50% faster because it saves a square-root of all the elements.

To time it, I used

t1 = time()
image = julia_set(mapping=mapping,
                  min_coordinate=-1.5 - 1j,
                  max_coordinate=1.5 + 1j,
                  iterations_count = 255,
                  width=1920,
                  height=1080)
print(time() - t1)

Before: 3.16s After 2.21s