The most likely competitor of Project Morpheus is the Oculus Rift VR headset, which was released in the form of a development kit in April 2013. The Rift initially received funding through a Kickstarter campaign in 2012, to the tune of nearly 10 times its initial goal. Facebook recently purchased the company for around $2 billion. The initial development kit had a 720p display and was missing positional tracking, but the new Oculus Rift Development Kit 2 (DK2) has specs more in line with Morpheus. The updated Rift includes a low-persistence 1080p resolution OLED display (slightly better than the Morpheus LCD display) and allows for a 100-degree field of view (down from the previous dev kit's 110-degree fov). For tracking, it comes with a camera and integrates a near infrared CMOS sensor, gyroscope, accelerometer and magnetometer. Its refresh rate is similar to that of Project Morpheus. Oculus Rift also includes a built in latency tester to help developers monitor and lower latency.
The Oculus Rift DK2 is $350 and is available for pre-order as of this writing, which puts it way ahead of Project Morpheus as far as availability goes. There isn't currently a projected release date for the consumer model, but tens of thousands of dev kits have already been shipped and several games are already available for the Rift, with many more in the works. Incidentally, "EVE: Valkyrie," which has been used to demonstrate Project Morpheus, was initially created for Oculus Rift. Right now there are only a few Project Morpheus prototypes, but more will reportedly be produced and made available to hundreds of developers as a dev kit, along with a software development kit (SDK) that's in the works.
Other Rift advantages are that it's somewhat platform independent (although right now it only works on Windows, Mac and Linux machines, no game consoles) and has open source software and hardware. Rift's software development kit (SDK) is available for free.
Valve, the software company famous for creating the games "Half-Life" and "Portal," created its own prototype VR headset and demonstrated it to a handful of developers at their Steam Dev Days conference in January 2014. The prototype was reportedly more impressive than the first Oculus Rift dev model. However, the company has no plans to release a consumer VR headset. They've even worked with Oculus to improve its product.
Two other VR headsets debuted at GDC 2014: the Sulon Cortex and the Seebright. However, both use your own cell phone as the display and content delivery system, so they aren't quite in the same league as Oculus and Morpheus.
So Oculus and PlayStation may be the only games in town for a while for fully immersive 3D VR unless other major players jump in. Project Morpheus has the advantage of being developed by a major company for a mainstream gaming console that as of this writing has already sold over 6 million units in the short time it has been out. The existence of PlayStation Move also gives them a readymade VR controller. But Oculus VR's purchase by Facebook puts a lot of weight behind the Rift, too. Both companies, however, seem to be viewing the other's fortunes as validation of VR as a viable technology rather than worrisome competition.