JavaScript, 284 283 281 277 272 267 265 263 240 228 227 226 bytes
currenty (2019-10-05) this is shortest js solution (last one has 285 bytes)
Expanded
// Fractal calculations for i-th pixel c=256; M=i=>{ j=x=y=0; while(x*x+y*y<4&&++j-c) [x,y]=[x*x-y*y+i%c/64-2,2*x*y+i/c/64-2]; return c+2*j // 4 digit num -> interpreted as base64 chars -> 3 bytes RGB }; // gen image base64 string for(p=k='';++k<c*c;)p+=7*M(k); // draw pixels in 256x256 BMP image document.write(`<img src=data:;base64,Qk02AAMAAAAAADYAAAAoAAAAAAEAAAABAAABABg${'A'.repeat(33)+p}>`)You can change numbers in return statement to change colors
c=256;M=i=>{j=x=y=0;while(x*x+y*y<4&&++j-c)[x,y]=[x*x-y*y+i%c/64-2,2*x*y+i/c/64-2];return 3*c-j};for(p=k='';++k<c*c;)p+=8*M(k) document.write(`<img src=data:;base64,Qk02AAMAAAAAADYAAAAoAAAAAAEAAAABAAABABg${'A'.repeat(33)+p}>`)