JavaScript, 284 283 ...(11 improvements)... 208 190 bytes
currenty (2019-10-05) this is shortest js solution (the shortest one before this has 285 bytes)
Expanded
// Fractal calculations c=512; // p contains 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+=9*c+9*j) 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${p}>`)After small constans changes
c=512;for(p=i='';j=x=y=0,++i<c*c;p+=2*c+9*j)while(x*x+y*y<4&&++j-c)[x,y]=[x*x-y*y+i%c/128-2,2*x*y+i/c/128-2] document.write(`<img src=data:;base64,Qk0bAAwAAAAAABsAAAAMAAAAAAIAAgEAGAAA${p}>`)