JavaScript, HQ: <s>228 226 210</s> 208 bytes
-
<sup>(Low-Quality: <s>284 283 281 277 272 267 265 263 240 228 227 226</s>)</sup>

<sup>currenty (2019-10-05) this is shortest js solution (last one has [285](https://codegolf.stackexchange.com/a/24080/88163) bytes)</sup>

[![enter image description here][1]][1]

Expanded

<!-- begin snippet: js hide: false console: true babel: false -->

<!-- language: lang-js -->

 // 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,Qk02AAwAAAAAADYAAAAoAAAAAAIAAAACAAABABg${'A'.repeat(33)+p}>`)

<!-- end snippet -->

After small constans changes

<!-- begin snippet: js hide: true console: true babel: false -->

<!-- language: lang-js -->

 c=512;for(p=i='';j=x=y=0,++i<c*c;p+=5*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,Qk02AAwAAAAAADYAAAAoAAAAAAIAAAACAAABABg${'A'.repeat(33)+p}>`)

<!-- end snippet -->


 [1]: https://i.sstatic.net/2hDcR.png