Farmington Bank is an old bank with a new branch in an old bank building. It plans to open its new branch on Elm Street in West Springfield today, Tuesday. Founded in 1851, the Connecticut-based bank, with headquarters in Farmington, started hiring staff and building business in Greater Springfield about a year ago. The location, 85 Elm St. at the corner of Elm and Central streets, was most recently a Bank of America location but was for many years a Third National Bank. It was built as the West Springfield Trust Co. in 1921.