Well the correct answer is both! If you could be very broad and very deep, and know everything wouldn't you? Alas though, even geniuses have their limits. So we have to chose what skills we are going to grow vs those we're going to sacrifice.
Generally it boils down to how much of a commodity you're going to be.
With a lot of breadth, you will always be marketable with a large skillset. You have more opportunities. However, as your skills are mostly surface skills, you will be more commoditized. That is to say, with more supply, eventually the salaries drive lower as the odds increase there's someone who can write that iPhone app for $30/hour where you want to charge $32/hour. The extreme example of "breadth" is manual labor which everyone can do and is very heavily commoditized.
With depth, the opportunities are scarcer. However, with deep knowledge you are competing in a smaller workforce. You'll likely be able to demand more $$ for your work as there are only a few people with those skills. You're not a commodity. The ultimate example of all-depth is someone with a PhD. You are so focused on a single problem that somebody may pay you through the nose to help solve that problem. You may, however, have problems finding the handful of people desperate to solve that subproblem. Depth can also be more brittle, the problem you're solving may pretty much be solved one day, making your job obsolete.
You have to find what balance is most appropriate to you between "manual laborer" and "PhD student". Software Engineer is already some level of depth as obviously not everyone can do it. But do you want to be deeper than that? I'd take some things into account:
Is my local area's job market sufficient such that there are multiple jobs at my desired level of "depth". Can I find another job in my chosen subfield easily? Or will it take a lot of work?
Do I want to stay focussed on the same problem my whole career, getting a very deep understanding and learning multiple ways to solve that problem, or do I want to always be learning new, disparate things even if its only surface level? Where do I want to be in that spectrum?
Do I want to command a higher salary, even if it means taking longer to find the job that best fits me? Or will I be ok on a smaller salary but more stability?
My Advice? Take a hedging strategy. Treat your knowledge like an investment. Get some depth in a promising field, but don't lose focus on learning other, sometimes completely unrelated things. Invest 60% depth and 40% breadth or 40/60, not 100% one or the other. Technology is way too fast moving to put all your eggs in one basket. But its also stable enough that many problems do stay around and linger. You want to not get too comfortable in one cushy, well paying gig, only to have that problem become obsolete with nothing to fall back on.