When working with a conventional word processor, it is often necessary to have several windows open at once (e.g. one for the main text, another for ideas or notes, etc). Constantly switching between them quickly becomes cumbersome and inefficient. Wiki's capacity to capture in text the kinds of intuitive and non-linear connections that the mind makes (see Wiki:WikiWordsAreConcepts) makes them ideal for both research and writing. Structure evolves with the content and vice-versa.
A WikiWordProcessor is much more than a PIM (see WikiAsPim), it is a notepad, word processor and database all-in-one. Over time, a database complete with emerging patterns and articles develops out of one's own notes and ideas. This database is accessible by a variety of means, including a front page, index, category system (see Wiki:ReverseIndex), and search engine. A WikiWordProcessor should be able to convert the wiki text into multiple formats including doc, pdf, html, xml, etc.
The following personal wikis come close to the functionality of a WikiWordProcessor:
See also WikiAsNotepad