Twitter will now use notification functionality in desktop browsers such as Firefox and Chrome to inform its users when they have received a direct message. Users must specify an opt-in for this.
Twitter announced the news via Twitter. Although the company reports that everyone can apply the opt-in, the link they gave leads to the the messages page, but it does not always show the option to enable notifications. Probably the function is somewhat gradually rolled out and sooner or later all users can use the notifications.
— Twitter (@twitter) August 24, 2015
Earlier this month, the company also announced that it raised the limit of 140 characters on DM’s. Thus, the company aims are to encourage users to use the social media for personal communication, which probably is also the motivation behind this addition.