Maybe in another day or two when a few more anime have been nominated I'll add a poll with them to this thread. That oughta be fun.
Meanwhile, my nominations for Best Anime of 2004 are Fantastic Children and Otogizoushi. They are both thought provoking and beautiful shows with striking animation, an involving plot and interesting characters.
PS. To keep the playing field level, I am only going to be including TV series in the poll, not movies or OVAs.
Also, I will be counting series that started during Winter 2003/2004 and ending with shows that started in Fall 2004. This is because the later shows will be too late to judge the few episodes seen before the end of the year. (Winter season begins 12/26) If you have any doubts about when a show started you should probably check ANN Encyclopedia
People should also be aware that Fullmetal Alchemist began in October '03 so it started in the Fall season. Therefore it is ineligible for this poll as it would have been on the "Best anime of 2003" poll if there was such a thread.