It kind of depends on what you hope to do in terms of playing jazz. In this forum, when people say they play jazz, they mostly mean playing solo arrangements of the heads of songbook standards. For this, it doesn't really matter what tuning you use.
If you want to play improvised solos and play with other instrumentalists (i.e. actually play jazz), I would highly recommend you play in C because you will end up playing in lots of flat keys (F, Bb, Eb, Ab, Db) and very few sharp keys. C tuning gives you more options in terms of voicings with open strings here. More generally, it is important to commit to one type of tuning so you can learn the notes of the fretboard thoroughly, which will absolutely help you to improvise fluently.