Mathew Rose is a character from the Alternate Universe and the first known individual to be freed and revived from Amber containment. He is currently living in hiding with his family since his existence would be a threat to Fringe and their use of the Amber due to the lie that people die when exposed, and the secret truth that they are needed to keep the Amber strong.
Matthew was accidently imprisoned in Amber when he went to warn his twin brother Joshua Rose who was robbing a bank that the Fringe Division was on to activities. Since his brother's technique to rob banks was causing micro breaches Fringe sent an Amber Division to contain the incident. While Joshua was successfully escaped Mathew became trapped in the Amber with Fringe and the public assuming he was his brother.
For the next four years he remained imprisoned while his brother, who was posing as him to avoid suspition, made plans to free him. In that time he experienced first hand the cruelty and inhumanity of the Fringe protocols. Not only was he still alive but his mind was conscious and trapped in his last terrified thoughts before imprisonment.
"Do you know what it was like for him? In the amber? Did he tell you? Well he told me. He was aware of every moment. His mind was stuck in the last thought he had before he was trapped. This horrible feeling of fear... and loss." Danielle describing her husband's imprisonment.
Four years later, his brother Joshua and his accomplice Mark Wilner broke into an Amber quarentine zone in Franklyn Street Station to retrieve Matthew from the Amber, and managed to revive him with several resuscitation techniques. Because of the breach new Amber gas was released to reseal the area. Unfortunately, Mark couldn't get away in time while both brothers made their escape.
His brother helps him return home to his family, where he recovers from a weakened state with blurred vision and a sensitivity to light; a side effect of his imprisonment and his bodies readjustment to the outside.
Olivia suspects the brothers of the identity switch and continues to pursue that line of investigation. After finding the blueprints of Riverdale Mutual he realizes that his brother is planning to rob another bank. Matthew goes after him again to convince him not to pursue his plan, and learns that Joshua is trying to redeem himself by getting caught, so that he can finally live a peaceful life with his wife Danielle and his two kids. After they quarantine the zone the Fringe team learns that they had one casualty: Joshua Rose.
In the end of the episode, Olivia shows up at his door step and tricks Matthew into giving her a skin sample and tells him that she knows his real identity and she could prove it. However, after seeing Rose's son at the door she decides against it and informs him that the investigation is officially closed.