Are there any other graphic systems that attempt to be as complete as the International Phonetic Alphabet?
The Finno-Ugric Transcription, a.k.a. the Uralic Phonetic Alphabet is comparably complete to IPA and much more regular and flexible.
I hope that this thread is not too dead, but I'd like to add IKPA - International Korean Phonetic Alphabet. It's based on Hangeul, so is mostly featural.
There is the Universal Phonetic Alphabet. I tried to post this some time ago, but couldn't find the link. Here it is; https://omniglot.com/conscripts/upa.htm
Visible Speech (1869) by Aleander Meville Bell and Standard Alphabet (1855) by Lepsius, whom Bell fit his characters on top of. Both systems are fully capable of expressing all of the phonemic distinctions that IPA does. If you would like primarily Latin based with Greek characters to supplement I would recommend Standard Alphabet. If you would like a character system that attempts to symbolize the organs used, Visible Speech does that. The Visible Speech vowel system is actually more complete than IPA, having 36 vowels instead of 28. Lepsius demonstrates how the Standard Alphabet maps to an exhaustive list of world languages at the end of his book. Bell has some examples of his system used in German, French, Cockney, American, and Gaelic. Since Bell's systems is based on Lepsius the mapping to world languages applies equally to his.
I have created a Unicode phonetic alphabet that I personally think is even better than the IPA.
It is the result of many hours of work on my behalf.
You can find the documentation on how to use it here: https://github.com/SalviaSage/Translingual-Phonetic-Alphabet
I update this page as I develop the alphabet further.
Here's another option: the Musa Alphabet. It's not based on the Roman alphabet at all, which leads to less confusion and clearer features. You can read all about it at www.musa.bet. And here's a page where Musa is directly compared to the IPA: musa.bet/ipa.