According to this analysis on CrimeTV.com:
The Massachusetts State House is featured in the film as a symbol of
Colin Sullivan's ambition.
This view is echoed in the Pacific Northwest Inlander article "Bloody Entanglements":
Damon manages to convey an astounding amount of information in scenes
where he doesn't even speak. From the way he sits and stares at the
Massachusetts State House dome, we get that he has political
ambitions.
There is no actual apartment in Boston with that view, however. According to imdb trivia:
The view of the Massachusetts State House was an effect shot from the
roof of Suffolk University, which is the law school where Sullivan
says he is taking night classes.
