JavaScript, 284 283 ...(12 improvements)... 191 190 188 199 bytes
currenty (2019-10-05) this is shortest js solution (the shortest one before this has 285 bytes)
Expanded
// Fractal calculations c=512; // p has pixels, p+= allways join 4 digit number implicit casted to string // 4 char string interpred as base64 gives 3bytes = 1 RGB pixel for( p=i=''; j=x=y=0,++i<=c*c; p+= j<c ? 9*c+9*j : 'AAAA' ) while( x*x+y*y<4 && ++j-c ) [x,y] = [ x*x-y*y+i%c/128-2, 2*x*y+i/c/128-2 ]; // draw pixels in 512x512 BMP base64 image document.write("<img src=data:;base64,Qk0bAAwAAAAAABsAAAAMAAAAAAIAAgEAGAAA"+p)After small constans changes
c=512;for(p=i='';j=x=y=0,++i<=c*c;p+=j<c?2*c+9*j:'AAAA')while(x*x+y*y<4&&++j-c)[x,y]=[x*x-y*y+i%c/128-2,2*x*y+i/c/128-2] document.write("<img src=data:;base64,Qk0bAAwAAAAAABsAAAAMAAAAAAIAAgEAGAAA"+p)