The alien computer has an interesting history with this series. Originally I wasn't going to use it/him at all, since he is arguably destroyed in Revenge of the Fly. Later I decided that I wanted him to still be around, but the plan was for Baxter to find him and for him to be with Baxter either only for one story to have closure or possibly indefinitely.
Sometime during the writing of Exit the Fly, I came up with the idea that Barney would discover him instead. He was last seen on the Technodrome, so very possibly he was still there. Barney discovered him at the end of that first story and he decided to make friends with Barney as he had Baxter.
Over the course of the series, the computer (whose designation is eventually revealed as V.I.N.CENT-2152) becomes Barney's conscience and confidant. It is through Vincent that the readers see into Barney's heart, instead of through monologues as was the case in the first story. Barney tells Vincent everything: the missions, what he thinks of them, his conflicting feelings about Baxter, and the depths of his self-hatred. Barney comes to love Vincent as his only friend and vows to take him along if he ever leaves Shredder and Krang's employ.
For a while Barney wants to selfishly keep Vincent for himself and doesn't tell Baxter, but Baxter accidentally discovers that Vincent is alive and is furious and hurt that Barney didn't tell him. Independent of that, Barney finally decides to tell Baxter, and from that point on, Vincent has many interactions with Baxter as well as Barney.
Shortly after Vincent's reunion with Baxter, he reveals that he has been lying to Barney all this time. Contrary to how he behaved on their initial meeting on the Technodrome, he knows all about Barney and has even met him before. But due to the dark circumstances of that meeting, which resulted in Baxter and Barney's memories of the incident being wiped for their own good, Vincent lied and pretended not to know Barney so as not to trigger the blocked memories.
Vincent later reveals that he didn't like Barney in the past and even encouraged Baxter to take revenge on him during the incident that was later temporarily erased from the brothers' memories. But when Barney ended up hurt worse than Baxter had even realized would happen and Baxter had a grief-stricken meltdown over it, Vincent realized that Baxter still loved his brother so much that even while his humanity was largely failing him, he couldn't bear to hurt Barney. He was then instrumental in helping Baxter reverse the damage.
Although initially angry at being lied to, Barney understands why Vincent did it and later is stunned and overwhelmed that Vincent has gone from wanting to take revenge on him to adoring him as much as he does Baxter. Vincent has seen the good that Barney can't see in himself and Barney has made a friend for life.
Barney eventually builds a custom purple laptop into which he places the alien computer motherboard, as he wants Vincent to be whole again. From that point, Vincent goes with Barney on the missions instead of staying behind in the laboratory.
In addition to Vincent helping Barney's growth, Barney has in turn helped with Vincent's. When he becomes friends with Barney and is faced with Barney's hatred towards Baxter, he can no longer encourage revenge. Instead he becomes the mediator between the brothers, listening to Barney's feelings and trying to help him sort through them until he can put the hatred aside and embrace the love he has for Baxter deep down.
In some ways, Vincent may have bonded closer with Barney than he did with Baxter, as Barney is of sound mind and Vincent can carry on intelligent conversations with him, whereas with poor Baxter, his mind was falling apart when Vincent knew him. But Vincent loves them both and is greatly enjoying getting to know a sane Baxter. His ultimate wish is for all of them to live together as a family, something Barney doubts will ever happen. In the meantime, Vincent chooses to stay with Barney even when he has the chance to go with Baxter, as he realizes that Baxter has a strong group of friends to turn to while Barney has no one but him.
Even as Barney continues to grow and develop and actually begins to cultivate a brotherly relationship with Baxter, it is unlikely that they will ever be as close as Baxter longed for. Barney's closest confidant and conscience will likely remain Vincent.