Tsubasa Chronicle 2nd Season

Both series involves time and dimension travel. Also, the male protagonist in both series travels and battles against his other self to save the girl that he loves. 

Both animes belongs to Clamp and presents the same characters in different universes. Both plots are essentially romantic, but they have many cool magical battles too! 

Both series are a crossovers from different series with the main male lead chasing after the female lead with a strong cast of characters. 

If your one who likes to listen to good soundtrack along with the anime, then both of these animes have what your looking for! ^^ The soundtracks in both, i would give a 10!  

Fushigi Yuugi and Tsubasa Chronicles, for me they have their similarities. It tells that in Fushigi Yuugi, Miaka Yuuki sucked into an ancient Chinese novel titled "The Universe of the Four Gods",sees that she can travel in other world because of the book. Miaka become Priestess of Suzaku, as the Priestess of Suzaku her duty is to gather the Seven Warriors or Seven Seishi of Suzaku, who are to be her protectors. while... in Tsubasa Chronicles, the similarities of the story between the two is when Princess Sakura lost her feathers that contains her memories. Syaoran, who is her protector, travels to different world in search of the feathers of Princess Sakura in able to return her memories.  read more 

Another anime by CLAMP, it is a cross-over anime featuring many characters from other CLAMP works, X/1999 included 

The same as Tsubasa Chronicles season 1, the main hero isn't aware of what happened during his childhood, because of an event he does not remember.