Is it possible to make laser beams visible midair without smoke?

It is possible to hit spots in the air with laser pulses from multiple directions in such a way that air molecules in that spot become ionized and emit light, see the technology discussed in this article, along with this demonstration video.

And if you just want a 2D screen rather than a 3D display, then of course you can also just use lasers to project it on a surface, like the "laser projection virtual keyboard" here.

Not really; you need to have the laser light pass through particles in a medium. Laser light is made of photons; in order to see the laser, photons must be reflected off of a something to your eyes. You cannot otherwise "see" a photon because photons don't interact via the electromagnetic force - in other words, photons don't emit photons. To see the laser beam, you have to see photons reflected off of something, and so you have to have it passing through some medium for this to work. Smoke works well, as does anything that's thick and fills the air. Normal air does not work well. So technically you could use something besides smoke, but you need to have the beam pass through some medium.

I recently saw a video of a demonstration by a Japanese researcher who came up with a method that used a pair of high-powered (presumably) infrared laser beams that, where they intersected, heated the air enough to turn it into plasma, creating a pulse of white light. It works, but it's slow, low-resolution, & requires staggering amounts of power. If you come up with a better method that actually works, don't tell anybody about it until you've patented it. :)