Has anyone thought of inventing a unified alphabet?

Yes. It's well known, and is called the International Phonetic Alphabet. You can read about it here: http://en.wikipedia.org/wiki/Ipa

It uses one letter (along with diacritics) for every sound used in human languages, and it is an internationally recognised standard tool. Linguists use it to represent pronunciations. For example, the word "speak" is represented in a "broad transcription" as /spik/: one letter for every phoneme.

The alphabet can also be used to emphasise the phonetic details of the pronunciation. Although most speakers are unaware of this, the /p/ in /spik/ is pronounced differently from the /p/ in /pik/ ("peak"). This difference can be represented using the IPA, in what is called a "narrow transcription". This makes the representation more complicated, but it makes detailed comparisons easier.

The IPA is also used as the standard tool on Wikipedia for representing pronunciations, and in many dictionaries.

No everyday alphabet of any language uses its symbols exactly as the same symbols are used in the IPA, but some come close, and some more unusual symbols from the IPA have been incorporated into the alphabets of some languages. It would be difficult to use any phonemically consistent alphabet to represent a language like English, however, because there's so much variation in pronunciation.

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Additional note: the Latin alphabet is not in fact phonetic. At best there was a pretty good letter-to-phoneme correspondence when it was used to represent Latin (which would make it almost phonemic, not phonetic). However, the correspondence was not perfect, and the degree to which there is such a correspondence for modern languages that use the Latin alphabet is very variable.

Almost impossible. The Latin alphabet is phonetic, but many aren't.