They have some goldfish together.
Costume Drama's including in The Life Aquatic and in a remake of The Creature from the Black Lagoon she is an actress of noted stature.
For example, while working on Wes Anderson's The Life Aquatic she ended up what started out at first as a somewhat friendly debate but ended up an aggressive round of fisticuffs with Bill Murray knocking him down before straddling and pounding him to death and then eating him. Eyewitness Bud Cort described Cate as "a pure animal" and said, "She could really move despite the pregnancy-simulating top on—if anything she used it to her advantage."
A furious Wes Anderson told Cate Blanchett to find a replacement for the finishing shots although he decided not to kill her because she was such a beautiful animal. She successfully created a Prosthetic Murray Suit which she wore and performed in as Steve Zissou. This is why Cate Blanchett is "uncredited" for co-playing the part.
Wes Anderson was edgy about the replacement Murray being convincing. However, many critics agreed that Bill Murray's greatest moments in the film were the ones in fact played by Cate Blanchett. Cate Blanchett has since been getting work as the late Mr. Murray to keep up the idea of Bill Murray being alive and is currently working on Ghostbusters 3, ironically as the ghost of Bill Murray's character, Dr. Peter Venkman.