Written Quechua has seen a number of paradigms in its history and it's and interesting case study for those interested in written standards.
The Inca would have been the first civilization to write Quechua down, but sadly it seems they never devised a system of writing. They left no discernible alphabet, abjab, ideogram (which is more of a quasi-writing system), or even hieroglyphics like those left by other prominent pre-Columbian civilizations, like that of the Maya. There is an interesting theory about how the Inca may have written which involves their famous quipu, a system to record numbers using knotted cords. It's an strange story involving snooty old families, ancient societies and artifacts, and Jesuits! I'm not going to bore you with what I don't really know much about. You can go here to learn more about how quipu were used and the theory on "literary quipu." I'd also suggest merely googling "literary quipu."
During and after the Spanish Conquests, Quechua had been written in the Latin alphabet. The written convention of the time was based off old Spanish orthography with a particular twist for writing Quechua. It had five vowel letters: a, e, i, o, and u. It also had a "hu" digraph which represented the labio-velar approximant, [w]. Another important feature to note were the letters "c/q(u)" which were used interchangeably for the velar and uvular plosives and did not distinguish between the three varieties of said sounds ([t], [t͡ʃ],[k],and [q], all have "plain," "aspirated," and "ejective" varieties that the old orthography failed to denote explicitly). This lead to a messy written standard with no "audio-visual isomorphism " (as a lojbanist would put it), but this would not be new to the world. Just look at what you're reading right now.
This hodgepodge of an alphabet was used for centuries until Peru made a standardized and modernized alphabet in 1975 and improved upon it in 1985. This new standard threw out the "hu" for a plain ol' "w" and shed the two extra vowels that they felt they didn't need: "e" & "o." It also distinguished between the different forms of [t], [t͡ʃ],[k],and [q] with "t," "ch," "k," and "q" being the plain forms respectively. You add and "h" to the end of these to represent the aspirated forms and an apostrophe to denote the ejective forms. This confused me a little at first as an Anglophone since "th" represents an aspirated voicless velar plosive as opposed to the labio-dental fricative I'm so used to, but the system is certainly an improvement on the old one. You may think everyone who speaks Quechua would embrace a new and improved upgraded system such as this, but you'd then be surprised to know that this is the center of a controversy and I believe I can explain why.
The new written standard is much more simple than the old forms, but there's been some backlash resulting in people supporting all kinds of standards and compromises. "Why would people fight progress?" you might ask. "This new system is great! They should just accept it," you might also say. But to play the devil's advocate, I posit this. What if English were to undergo a similar shift? What if were rid our writing of clumsy digraphs such as "th," "ch" and "ng" and replaced them with "þ" and "ð" for "th" like the Icelanders and "ч" for "ch" like the Slavs and "ñ" for "ng" like the Spaniards. We"ll also need to rid ourselves of suч clumzy practices as usiñ tonz of letterz for one vowel sound. We'll stick to "i" for "ee," "o" for "o," "u" for "oo," etc. Gydnes, Ai laik ðes nuw sestem myч mor alredi. If you haven't pick up on it yet, improving upon an alphabet that's already been in use for centuries is difficult, confusing, and can split up a language. If English was written as it sounds, I wouldn't be able to read what some people with thick accents were saying and our languages would split! ( I intend to dedicate more to this idea in the future.) The sort of spelling change that has happened in Quechua would NOT fly with English writers and readers.
All that being said, there's a fundamental difference between Quechua and English, the number of literate speakers. It is true that a spelling reform like what has happened to Quechua would not work in a major world language like English since it is read by so many people, but while Quechua has eight to ten million speakers, relatively few are literate. That number is of course growing. Which makes now, or, rather, the 1980's, and opportune time to clean up the written standard. As the younger Quechua population grows, learns, reads, and writes the new standard will be accepted with backlash from only those who grew up with the older methods.
A detailed page about this same issue:
A short wikipedia article on the change in standards
The Omniglot page on written Quechua.