You only need a computer and some way of hearing the sound from BIAB (and preferably something better than built-in speakers). It gets more complex if you're wanting to record or play instruments along with BIAB though, are you wanting to do that as well?

As an alternative to studio monitors or better speakers, I'd suggest a good pair of headphones - preferably the larger "studio" kind that go over your ears. I personally use a pair of Audio Technica M50X's, but most studio headphones in the $50+ range will be good enough. Generally speaking, more expensive means better sound but also better comfort and less ear fatigue, and when you're working in audio professionally you don't want headphones that are uncomfortable or tiring to use. As well, some people will tell you that you'll need a better headphone amp to run your headphones, but in my experience that's not always the case. Some "professional" headphones require one, but usually they'll be a 600-ohm headphone. The Audio Technica's I mentioned don't require a headphone amp.

That said, I personally prefer monitors over headphones.

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