Here's a quick post because I think I finally found a better justification for C-extensions than "composers wrote low notes and I need a way to play them" and "they look cool" and my personal favorite, "because I can."
They can make any key a resonant key.
Anti-climatic, perhaps. But I'm convinced this is the best justification for having a C-extension. Having the means to crank out low notes is just a nice bonus.
I think the conventionally resonant bass keys are C, G, D, A, and E, give or take a couple of accidentals. These keys all have notes that occur as open strings or as strong harmonics on one or more open strings, so intonating them is easy and they sound full and speak quickly. Keys like Db, Ab, Eb, etc., are less resonant since the notes in the key often conflict with open strings. The extension can change that, however.
A Quick Note on Intonation
I think good intonation is more about achieving resonance with the bass than getting the tuner needle to hit "0" every time. This is a freeing, flexible way to approach intonation that can work wonders for sound production, but unfortunately it's less effective with keys that don't have many open strings.
Octaves, unisons, and fifths are the most resonant intervals on bass, so I check my intonation using those intervals (as open strings or harmonics) whenever possible. Here are a few examples:
Thirds are a good reference in some cases, too:
The examples above derive their resonance from open strings and their harmonics. That's great and incredibly helpful if you're playing something like the opening of Mozart 35...
...but not as great if you're playing something like this theme from Mozart 39:
The D's, G's, A's, F#'s, and B's in Mozart 35 all coincide with resonant open strings or harmonics, so the excerpt speaks well, sounds full, and is generally idiomatic to the bass. The Mozart 39 excerpt is still technically idiomatic, but the Eb's, Ab's, and Bb's create a lot of conflict with the same open strings and harmonics that make the first excerpt so resonant.
Thankfully said conflict has a simple solution: close the Eb extension gate.
Now, the "Eb string" provides harmonic reinforcement for the otherwise wolfy Eb's, Ab's, and Bb's. The bass speaks more easily and this gentle Eb melody flows almost as resonantly as one in D Major.
I hate playing in Db minor. It has nothing to do with the key signature or the double flats but rather the lack of resonance from Db and Ab, two decidedly important notes in the key. I have a bad wolf tone on my D string Ab, so I close the Db gate when I play a passage like this one from Rigoletto. It gets rid of the wolfiness (Ab is a strong overtone on the low Db) which in turn helps me bring out the long, pianississimo phrases. A bit of enharmonic E string resonance might be lost on the Cb's and Fb's, but it's worth the trade-off.
If there are no extension notes in the music, I often close the extension on a note that will provide lots of harmonic support (usually I, VI, or V), both in terms of harmonic function and literal overtones, to the music. I do this regardless of whether or not the key is one I would consider inherently "resonant." In A Major I might close the low D: the A overtones would benefit the resonance of the tonic and the low D would give me access to a nice, deep VI-V motion if needed.
This post was admittedly biased towards chromatic c-extensions (non-chromatic ones have their benefits, too), but I stand by what I say. Having access to a new world of resonance really is the best reason to get an extension.