I'm sorry, but I do not understand where all the praise is coming from. The answers are simply incorrect. And the explanation merely repeats the answers that the author thinks are correct, and then gives a link to a page that does not support these answers.
* "These types of indexes do not exist in SQL SERVER 2012". Indeed, this is a false statement. No argument here.
* "They reduce the index maintenance cost compared to index views". Not true. The link given states that index maintenance for a filtered index is less than that for a full-table nonclustered index (and even that is not necessarily true). There is no comparison made to indexed views. An indexed view that includes only a single row from a million row table will probably have lower maintenance cost than a filtered index that includes half of those rows.
* "This is one type of Clustered index". Indeed, this is a false statement. No argument here.
* "This is an optimized nonclustered index". Nope, nothing optimized about it. It is exactly the same as a "normal" nonclustered index, except that it includes less columns. Yes, I did see that these words are an exact copy/paste of Books Online - so that simply means that Books Online is wrong.
* "You can create a Filtered Index on a view". Indeed, this is a false statement. I though it would be possible, but I had never tried it so I decided to try it - and I got error message 10610, "... Filtered indexes are only supported on tables ..."
* "These indexes improve the plan quality during query compilation compared to index views". Rubbish. In comparison to full-table nonclustered indexes, a filtered index might, when designed and used correctly, improve plan quality (if it is being used at all - more on that in my next post). A well-designed indexed view may optimizer the plan even more. It all depends on what you are querying. An indexed view can make joined and aggregated data available directly from the index. A filtered index can never do that. The only benefit a filtered index has is that it contains less rows.