From poking around Wikipedia, it looks like the CM&StP had some Standard Oil and other New York ties on its Board of Directors, but was locally controlled until the Panic of 1893. (See http://www.psmre.org/hist-milw.htm
for a bit more on this.) According to http://en.wikipedia.org/wiki/Northern_Pacific_Railway
, Henry Villard and Brayton Ives (former head of the New York Stock Exchange) fought for control of the NP through 1896, along with three separate bankruptcy courts. In 1896, James Hill purchased his first interest in the Northern Pacific, and in the same year, J.P. Morgan was called in to right the financial ship. Eventually, they created Northern Securities to jointly hold the GN, NP and CB&Q.
So, the Milwaukee had three years in which it could have possibly purchased the Northern Pacific without major interference by Morgan. Major backing could conceivably have been obtained from CM&StP director and Standard Oil oligarch John Rockefeller, as well as several other major shareholders. However, none of the major Milwaukee Road shareholders had apparent ties to Villard, Ives or Morgan (until later), and it might have been difficult to break into any of those cliques. More importantly, the post-Panic Northern Pacific was a legal and financial minefield - three bankruptcy courts claimed jurisdiction over the reorganization process. It's very possible that even if the CM&StP had considered a bid, they might have decided against it due to the potential pitfalls, or the courts might have blocked such a transaction.
Having said that, the Milwaukee was still intensely profitable in the 1890s. Compared to the eventual $257 million (from Wikipedia) it cost to build the Puget Sound Extension, or even the $45 million original quote in 1901, the full Northern Pacific might have been a relative bargain with its land-grant lands, mineral and timber rights, and burgeoning population en route. I have no idea how a purchase could have been effected, with the stock tied up in bankruptcy court, but if J.P. Morgan found a way, John Rockefeller and company probably could have done the same. There wasn't really any parallel track structure yet, as the Milwaukee wouldn't really start building west of the Missouri until the 1900s, and the eventual cutoffs from St. Paul to Terry, St. Regis to Spokane, and Lind to Ellensburg would have had the backing of a much larger railroad without direct competition. The Milwaukee would have had to do some house-cleaning, as the Wikipedia article claims a fair amount of mismanagement within the NP hierarchy, but even that might have been easier than working from the ground up.
Very interesting what-if, CPF363. Hopefully I provided a bit more information for you to mull.
EDIT: If you're interested in another what-if situation, consider if the C&NW had put together an effective bid for the Northern Pacific. The Chicago and North Western doesn't appear to have had as powerful an investor base as the Milwaukee - at least, I can't tell much about their investor structure aside from what the C&NW Historical Society
says - but they had rail connections in many of the same places and a strong corporate earnings sheet.