Some people try to complete their PhD asap, others try to make major breakthroughs. I think these aims are mutually incompatible, one is going against the other.
What are pros/cons of each for somebody whose intended career is research in academia?
The answers should be general, but I myself am interested in pure mathematics.
If you are aiming for a career in mathematics at a research university, then trying to complete your Ph.D. as quickly as possible is a bad idea. In the U.S., you should plan on spending four or five years in graduate school. You should not graduate in three or fewer years unless you write an absolutely extraordinary thesis (or personal concerns force you to leave graduate school). Graduating in four years is reasonable, but most people are better off taking five. Even if you've reached a natural break point in your work, it can be valuable to stick around and work on something else for a year.
This may seem counterintuitive, but it's based on three principles that describe how junior candidates are evaluated in mathematics:
The job market is brutally competitive, and you need the strongest application you can put together.
No extra credit is given for graduating quickly. People who graduate in three or four years compete on the same footing as those who spend five. [However, there can be a penalty for taking six or more.]
Once you graduate, the clock starts ticking. If you take longer than average to find a tenure-track job, it will be held against you.
The net result is irrational: the job market judges it better to spend five years in grad school and then do a three-year postdoc than to spend three years in grad school and then five years as a postdoc. The former looks normal, while the latter looks like you graduated too quickly and then were unable to find a tenure-track job after your first postdoc (so what might be considered a compensating plus/minus pair turns into two negatives).
Five years in graduate school is something of a cut-off, for two reasons. Most grad schools discourage staying longer, because of limitations on space and funding, and spending six or more years starts to look like you were unable to complete your dissertation. However, up to five years, longer is usually better.
One way of thinking about this is to consider the question title:
Longer PhD with a deeper result vs a shorter PhD with a sufficient result
What constitutes a "sufficient result"? Sufficient to graduate is a weak condition, since getting a good postdoc is much more difficult than merely completing a dissertation. However, a good postdoc won't get your career off to nearly as strong a start as a great postdoc would. If another year in graduate school could make the difference between a good and a great postdoc, it may well be worth it. But what if your work is so wonderful that you're obviously going to get a great postdoc? At that point you should raise your ambitions and aim to get a Clay research fellowship.
There's an enormously high ceiling, and you never reach the point of being able to say "OK, I've done enough." Graduate school is not a matter of doing enough and seeing how long it takes you. Instead, it's a matter of taking the standard amount of time and seeing how much you can do.
There are several considerations regarding whether to graduate now. I'll go through them from most to least important (in my opinion).
Future job situation:
Job prospects this year vs next year: For some fields, getting a good job requires a student to have papers in advanced stages of the publishing pipeline, i.e. submitted is good, under second or third round of review is better, and accepted is best. If you have a great paper but it is not yet ready, waiting until the next year before you apply for jobs may allow you to have that great paper in a more advanced stage of the publishing pipeline which can't hurt your job prospects.
somebody whose intended career is research in academia?
I believe any PhD student who can
should graduate and enjoy the freedom of research. PhD is not a lifetime career. It is just the beginning. Longer or shorter PhD does not matter; what does matter is that you are an independent researcher when you defend.
others try to make major breakthroughs
This can be a trap because not everybody is able to make major breakthroughs. Besides not being smart enough, a major factor is the group of people you are surrounded by. It could easily take years for your cohort to catch up with the leaders in the field. If your project involves a major leap, consider the alternatives.
Evaluating what is do-able and the time involved is understanding your field, and an important part of a PhD. In fact, big companies have 'domain specialist' positions where your job is to review the value and effort involved in a project.
There is a category of people with a fast PhD who are successful because they understand what is relevant to pursue.
My supervisor used to tell me regularly that there is no such thing as a "PhD with bar" (that is some extra merit/credit/marker to say it was really good - it is just yes/no).
On the other hand several of my students have one university or professional association prizes for their theses, and there are opportunities for publishing a good thesis. A good thesis is something you will want to cite for the rest of your life. You don't want something that you will be ashamed of in a decade - whether or not you "pass".
I get students to decide on three innovations that if they succeed with any one of them would be sufficient for the PhD. Some may be dead ends, but most students do manage to hit two or three bulls eyes. But one is enough.
The questions really are: will your thesis get up? are you employable? could you get a research job? The way to be confident of this is to have multiple publications (my students tend to have 5 from their PhD, mostly conference, with a couple of journals - those who did a Masters will have some from them as well).
The other thing is that if you have a scholarship or a TA contract or other support, make the most of it, use all of it - don't submit early. It is far easier to publish during your PhD than once you are in a full time academic position preparing teaching notes for half a dozen different courses. So aim for publications first, building up your thesis as you go (either a resource from which you pull out publications, or a compilation made by putting together your papers). The papers related to a chapter should be identified in the opening and/or closing pages (e.g. with footnotes), and in some universities you can do a thesis by publication or include verbatim papers with a bit of glue text. But examiners don't really like this (they tend to read the same intro material over and over again), and I encourage a coherent readable monograph approach (which is then publishable which a collection of already published papers isn't). But one way or another if referees from a stack of reputable conference and journals think your work is publishable, how can they turn down the thesis that spawned them.
