The OP did say he's looking for a bookkeeping position, the financial industry is still one of the most staid industries because you are dealing with other people's money, so you will be scrutinized on appearance.
Been in business for over 35 years so you can take my advice for what's it worth to ya. Hired hundreds of people and managed thousands and know how many white collars think in my dealings with them over the years.
If you're starting out looking for a job, don't grow your hair, there's still a stigma in the office environment you are looking to work in. Once you get hired don't get confident and start growing your hair right away. Prove yourself as a valuable addition to the company and they need you, once you feel confident and accomplished then you can grow your hair, at that point you have proven yourself and any first impressions go out the window. I have worked with a few professionals with long hair and all have been in the workforce for a while and are excellent at what they do, I don't care about their hair.
But starting out now, you will be judged and first impressions mean alot as there is so much competition when looking for a job. Good luck.