Eric's answer is quite thorough, but it is missing an important point: in mathematics, everything tends to be related and intertwined. You might think that doing abstract algebra will get you far from analysis, but you would need to restrict your scope a lot more to do that, as there are many things relating the two, just to name a few:
- Group theory and group representation theory, which is very closely tied to harmonic analysis.
- Complex analysis is often useful whenever we deal with algebraically closed fields, both as a tool and as an illustration. Similarly with real analysis and real closed fields.
- Algebraic geometry is closely tied to analytic geometry.
- Topology (which you seem to include within analysis) is ubiquitous in mathematics, and that certainly does include abstract algebra.
I could probably name a few more, and I am neither an algebraist nor an analyst (though to be fair, I have had some education in both, quite a bit beyond the undergraduate courses).
There are branches of mathematics that could possibly get you further from analysis (some branches of mathematical logic, perhaps). That said, I can barely imagine how one can do serious mathematics without at least a modicum of topology.
Last but not least, in my opinion, really beautiful mathematics is the kind that (among other things) connects seemingly unrelated concepts and results in a way that is both unexpected and effective. By judging parts of mathematics as entirely uninteresting (especially such large parts as you seem to be indicating!), you severely limit your ability to even understand that and perceive that beauty (leaving you with a black box at best), not to mention making anything like that on your own.
Edit: I've just recalled a funny bit of somewhat related trivia: in the first chapter of Serge Lang's algebra (which is a canonical reference for abstract algebra, from basics up to quite advanced topics, and generally steers clear of analysis), Lang gives two concrete examples of monoids. The first one consists of the natural numbers with addition. The second one consists of... homeomorphism classes of compact connected surfaces with connected sum, indirectly referencing surgery theory and classification of compact surfaces.