Can I go bouldering alone?

Yes! Bouldering is one of those sports that you really don't need anyone with you to do it, you can sign up to a climbing gym and just start climbing.
It's a very friendly sport, most climbers are very helpful, open and willing to help others and climb with others, if of course that is what you want, if not then you can always just go and climb on your own.

It does help to climb with others, it helps when you have others to bounce ideas of off, or to learn from, but even if you want to go and climb alone as a hobby, a meditative practice, or what ever the reason, you can just go and do it.

If you are asking about outdoor bouldering, then the answer is no, you will need additional people with you to keep you safe and watch over you as you climb. Climbing gyms are different in this aspect as the routes are usually built much safer than the routes you may find outdoors, they are usually also shorter, so even if you fall, you don't fall from very height up. In gyms, you also fall on climbing mats, which are basically mattresses that soften your fall, whereas with outdoor bouldering you will need to bring enough crash pads to cover the area below you, which can add to the risk if not well-prepared.