Laser noob here. I don't know what the maximum range on a high powered laser is, nor a low-powered laser for that matter. Just wondering if this is at all possible. I mention four extremities because it would a combination of the four different laser readings of height. I guess it could be most useful over water, but things could get complicated if flying over rough regions with mountains and valleys, but could be helpful if use in conjunction with a map.
Yes, you could create an altimeter using a laser. And they exist; See Laser altimetry for aircraft and UAVs by Renishaw as one example.
Laser altimeters, however, have a significant disadvantage over radar altimeters: They can be obscured by weather, like clouds and rain.