An excuse to use the obscure .BY:
DT[, newval := .SD[[.BY[[1]]]], by=new]
col1 col2 col3 new newval
1: 1 4 55 col1 1
2: 2 3 44 col2 3
3: 3 34 35 col2 34
4: 4 44 87 col3 87
How it works. This splits the data into groups based on the strings in new. The value of the string for each group is stored in newname = .BY[[1]]. We use this string to select the corresponding column of .SD via .SD[[newname]]. .SD stands for Subset of Data.
Alternatives. get(.BY[[1]]) should work just as well in place of .SD[[.BY[[1]]]]. According to a benchmark run by @David, the two ways are equally fast.