Warning: SPOILERS for Thor: Ragnarok are ahead!
Thor has been around over a millennia in the Marvel Cinematic Universe, but he's undergone tremendous character development in just the last several years. He found his humility in the first Thor movie, he learned to work with other mighty beings in the Avengers movies, and he was wise enough not to take the Asgardian throne in Thor: The Dark World, realizing that he wasn't ready yet to be a ruler. Thor: Ragnarok, however, arguably marked Thor Odinson's biggest development yet, and it all started with the loss of his trusty hammer, Mjolnir. Now that Mjolnir is gone and Thor is more powerful than ever, it begs the question of if the God of Thunder will wield a new weapon when he returns next year in Avengers: Infinity War. For right now, the answer to that query appears to be yes, although it's unclear if that will stick.
As was shown in the Thor: Raganrok teaser trailer back in April, Hela instantly destroyed Mjolnir when our principal protagonist challenged the Asgardian goddess of death (though this shocking scene was shifted from a New York alleyway to a field in Norway for the finalized theatrical cut). Imprisoned on Sakaar soon after, Thor made due with the weapons provided by Grandmaster for gladiatorial combat, but as we've seen in Thor's earlier MCU appearances, those are but mere trinkets compared to what Mjolnir was capable of in battle.
Fast-forward to Thor: Ragnarok's climactic battle on Asgard, Thor was soon overpowered by Hela, losing an eye in the process (like father, like son). After suffering this injury, Thor had a vision of the now-deceased Odin, who told him that Asgard is a people, not a place. When Thor stated that he couldn't defeat Hela without Mjolnir, Odin reminded him that he's not the God of Hammers, but the God of Thunder, and that Mjolnir was only a conduit for his powers, not the source of them. With this realization, Thor was able to draw upon the full might of his thunder powers and even the playing field against Hela. It took having to free Surtur and allowing the Ragnarok prophecy to pass to defeat Hela (resulting in the destruction of Asgard after its people were evacuated), but now we have an eye-patched Thor who has full control of his abilities, making him even more powerful than Odin, according to the All-Father.
Which brings us to Avengers: Infinity War, where Thor will be among the many superheroes battling Thanos and the Black Order. So far the only thing we know specifically about Thor's involvement in Infinity War is that he will run into the Guardians of the Galaxy early on, as revealed in the D23 footage this past summer. However, regarding Thor's weaponry, back in February, Marvel posted an Avengers: Infinity War behind-the-scenes video, and one piece of concept art showed Thor fighting alongside Rocket Raccoon, only Thor looked like he was armed with an axe, as you can see below. This video was released months before the Thor: Ragnarok teaser trailer, so this was our first indication that Mjolnir wouldn't be around for much longer. There's also precedence for Thor using an axe, as he's been using Jarnbjorn in the comics ever since he was deemed unworthy and lost the ability to use Mjolnir following the Original Sin event.
So where does this leave Thor weapon-wise? Between his enhanced thunder abilities and natural super strength, it's not like he needs to use a weapon to tear through his enemies. Still, Thor is a warrior, and it's not like wielding an axe would make him any less formidable in battle. However Thor obtains this axe in Avengers: Infinity War, we expect he'll be able to do a fair amount of damage with it, but it remains to be seen whether this will be a temporary or permanent weapon. We also shouldn't discount the possibility that Mjolnir could somehow be reformed in the future. Given Thor's enhancement, him using it again would be more out of familiarity than necessity, so perhaps instead of returning to the God of Thunder, Mjolnir could find a new wielder, paving the way for the introduction of Beta Ray Bill, a female Thor or someone else considered worthy enough to possess the power of Thor (but not all of his power).