##Java - 505 405 324
Java, 505 405 324 bytes
Just a standard calculation, with golfitude now with extra golfitude.

Golfed:
import java.awt.*;class M{public static void main(String[]v){new Frame(){public void paint(Graphics g){for(int t,s,n=640,i=n*n;--i>0;g.setColor(new Color(s*820)),g.drawLine(i/n,i%n+28,i/n,i%n),setSize(n,668)){float c=4f/n,a=c*i/n-2,b=i%n*c-2,r=a,e=b,p;for(s=t=99;t-->0&&r*r+e*e<4;s=t,p=r*r-e*e+a,e=r*e*2+b,r=p);}}}.show();}} With line breaks:
import java.awt.*; class M{ public static void main(String[]v){ new Frame(){ public void paint(Graphics g){ for(int t,s,n=640,i=n*n;--i>0;g.setColor(new Color(s*820)),g.drawLine(i/n,i%n+28,i/n,i%n),setSize(n,668)){ float c=4f/n,a=c*i/n-2,b=i%n*c-2,r=a,e=b,p; for(s=t=99;t-->0&&r*r+e*e<4;s=t,p=r*r-e*e+a,e=r*e*2+b,r=p); } } }.show(); } } 