http://www.theverge.com/2015/7/6/8899807/instagram-1080p-pictures-photo-upload
The Verge recently made known the latest app versions of Instagram are uploading photo
I'm unsure whether it's possible to get a 1080p resolution image. But here is one way to get an image with 320x320 resolution.
1) Enter the username of the user in the following URL. Here I'm using simonsterstrength as username.
https://www.instagram.com/simonsterstrength/?__a=1
2) Search for "profile_pic_url_hd" and copy the URL and access the picture that IG calls it as HD.
Note: As of June-2020, the response from IG CDN is OK.