Sadly, this is not currently possible. My best workaround suggestions are to use Chrome profiles to sign into both accounts, or to share the boards you need to access with both accounts, so you can edit your work boards from your personal account and vice versa. Sadly, these are workarounds and not the ideal solution you propose. I would advise you to submit this as a feature request to Trello Staff through this link. It's a great feature idea :)
I wish I had a better answer for you, but I hope you're able to find a solution that works in the interim!
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