While you are still a scholar and doing research, a lot of it will be original in nature so it is quite sensible that you try and publish your work, at least some of it, while you are still doing your PhD. It is all the more important to do it now, at this stage as it is the time that you can look for support for publication from your supervisor and benefit from his guidance. In addition, scholars find it easier to accept rejections in their papers as compared to seasoned academicians who cannot accept and absorb rejection that easily. Which means that while you are a scholar, it is perfect timing to do as many publications as you can.
Well, having said all of that above, the PhD duration is the best time to try and make publications, how good an idea it is to have co-authored publications? When I talk of co-authored papers, of course there is a choice to have anybody as a co-author but if you have a supportive, willing and understanding supervisor then it makes perfect sense to team up with your supervisor to get a publication which meets very good standards.
In disciplines such as the sciences we commonly get to see multi authored papers and it is quite expected and obvious to find papers that are multi authored. If you do belong to this discipline, it is perfect to take the initiative and suggest to your supervisor to have co-authored papers.
If and when you want to suggest to your supervisor about co-authoring a paper with you, never go blank. You must go with some preparatory research with some impressive idea or a research question, outline plan so that when you approach, you have something to discuss and in addition your supervisor feels confident that when he co-authors with you, he can expect you to make sufficient and constructive contribution, whoever takes the lead in writing the paper, the role of each one should be clarified and segregated from the very beginning itself.
So though it is a good idea to co-author a paper with your supervisor and it surely will work in your advantage as it would help you to get into academic publishing but you must never initiate the process until you are well prepared so that clear ground rules for the cooperation can be expected. Good luck!!