How hard is it to succeed in research after an adviser's death midway through PhD? TL;DR: How hard would it be to finish out a Ph.D. and have an effective early career (as a pure mathematician) after my adviser has unexpectedly passed away? My department does not have any tenure-track faculty able to take on students in my field at the time, but two postdocs in my field have offered to help, and the department has offered to fly me to visit researchers in my field. I am currently in my fourth year at a U.S. university.

I read this similar question, but it seemed to address practical plans for going forward more than the costs and benefits of doing so. What I'm looking for is advice on how difficult it would be to have a successful career and/or finish the Ph.D. in a reasonable amount of time, in addition to talking about what options I have.

I am a pure mathematics Ph.D. student in my fourth year. My adviser recently passed away unexpectedly, and my department does not have anyone else in his field who is able to take on students. He was a great loss to all of us, as he was a deeply intuitive mentor as well as an excellent researcher. Before my adviser's passing, I had anticipated 1.5 to 2.5 more years to Ph.D.

In the next six months or so, my department has offered me the option of working with a couple of postdocs (in my field) at my current institution, as well as flying out to occasionally meet with more senior mathematicians at different institutions. It is unclear whether the department will hire another researcher in my field anytime soon, especially one who is not fresh out of a postdoc. The department is currently uncertain if there will be tenure-track faculty there who could advise me anytime before I graduate. The department tends to offer at least 6 years of funding to Ph.D. students.

I currently have enough material for roughly one paper, but I feel that I still need guidance to finish my Ph.D., and I felt like I was really benefiting from weekly/biweekly meetings with a single adviser. My adviser helped keep me confident and excited, as well as explaining difficult techniques and offering new approaches to proofs or counterexamples.

I have heard some people advise on transferring institutions at this point, though my department does not recommend that. In addition, some people have pointed out to me that an adviser is an important early-career research collaborator, to the extent that it is difficult to do good research early in a career without having them to collaborate with or ask questions of.

If my time to Ph.D. is going to be lengthened significantly, I would like to know that going forward, as it would help me make decisions about whether to transfer, etc. So my question is: how hard would it be to finish out a Ph.D. and have an effective early career without an adviser in the traditional sense?
In academia, as everywhere, life happens.
Fortunately, a lot of people understand this.

It will definitely be harder to succeed when your advisor has passed away than if your advisor had lived.  I suspect, however, that it will be much easier than if you had a falling out with your advisor, which has been addressed in a number of different questions on this site.

It sounds like your institution is doing the best it can to offer you options for getting new advising, and that those options are not so bad.  It is also the case that you are reaching the point in your program where you would in any case need to start reaching out to build strong relationships outside of your home institution.  Where your advisor might once have helped bootstrap your connections, you have instead your institution's offer to help you meet with other mathematicians.  This is a big juncture in your life and career, and everything changes.  Despite the fact that the original quote is false, a crisis really is both danger and opportunity.

Also, make sure to take care of yourself.  You just had somebody close to you die, and that might affect you in a lot of ways, including interfering with your ability to work.  If you aren't already doing so, I would suggest proactively starting to see a counselor to help you navigate any psychological challenges that may come up: even if you don't feel you are having troubles now, they may start and it is better to catch them early, rather than having troubles that you might attribute to your new advising and then realizing six months down the road that it was actually your reaction to the death.