Actually, there's really not much point to you being notified that there's an accepted answer. All the big green checkmark means is "this is the best answer that helped the original question asker", and it by no means should preclude you, or anyone else for that matter, from posting your own answer, upvoting other helpful answers, or using the information posted in a non-accepted answer to solve a problem you're facing.
Additionally, many times when I use Stack Overflow during my work, the answers that help me the most oftentimes aren't the accepted answer. Every person's problem will be different.
With that said, you might check out Stack Apps, as there may very well be a user who used the Stack Exchange API to build an app that does something similar to what you're asking for.
Check out StackStalker. It looks promising! Hope this helps!
Also, if there is a question you see that is unanswered, you can always post a bounty on the question, which will get it more attention and increase the likelihood that it will be answered. However, before posting a bounty, be sure to edit the question to clarify it. This will not only make it easier to understand, but also it will bump it to the top of the active main page.