Just leave them and let them grow with the "mother" bulb. In a year or two, you will have a large enough bulb to plant on its own. Many bulbs create a nice display if they remain together in a pot. If you're lucky, you can end up with lots of flowers at once!

Do I just plant the bulb at the normal depth; with the green side shoots below the soil?

I plant all of my amaryllis bulbs only deep enough to cover the baseplate of roots. Here's one of mine that has sent out two side shoots two years ago, and those side growths are still in the same pot. I just went and checked on them and the two side bulbs are roughly half the size of the main one. I also noticed the pot they're in is beginning to warp from the bulbs running out of room, so I guess I'll repot them in the spring before they start growing again. Plus, the smaller ones are much lower in the soil and growing at an angle. I plan on separating them from each other, but still keeping them growing together, because I don't really need three separate pots of the same plant. I think they look better growing together anyways.