JavaScript, HQ: 228 226 210 bytes
(Low-Quality: 284 283 281 277 272 267 265 263 240 228 227 226)
currenty (2019-10-05) this is shortest js solution (last one has 285 bytes)
Expanded
// Fractal calculations for i-th pixel c=512; for(p=i='';++i<c*c;){ j=x=y=0; 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]; p+=9*c+9*j // p+= allways 4 digit number implicit cast to string } // 4 char string interpred as base64 gives 3bytes = 1 RGB pixel // draw pixels in 512x512 BMP image document.write(`<img src=data:;base64,Qk02AAwAAAAAADYAAAAoAAAAAAIAAAACAAABABg${'A'.repeat(33)+p}>`)After small constans changes
c=512;for(p=i='';++i<c*c;){j=x=y=0;while(x*x+y*y<4&&++j-c)[x,y]=[x*x-y*y+i%c/128-2,2*x*y+i/c/128-2];p+=5*c+9*j}document.write(`<img src=data:;base64,Qk02AAwAAAAAADYAAAAoAAAAAAIAAAACAAABABg${'A'.repeat(33)+p}>`)