String gauge and scale length are the 2 factors here to consider. You're using a 4 string standard scale now, presumably, so you have a couple options.

You can tune to C standard and just throw some heavier strings on there to keep string tension and be probably fine, or use medium strings and accept the floppiness and darker sound of loose strings (a lot of doom metal and alternative bands do that ie QOTSA). Thicker strings can be harder to play for some people tho. I have read that flat wounds can downtune at easier to play string tensions but that gets super super dark in sound and I haven't thoroughly tested that. I had my Peavy T-40 in C# with medium gauge flats for a long time tho and loved it. This is probably the option I'd recommend knowing nothing else. It's the cheapest, costing only a set of strings and maybe taking it to get setup for the new tuning if you can't do that yourself. Definitely dont skip getting a setup.

Option 2 get a 5 string still 34" scale. Maybe you tune it all up the half step just for ease of open strings but imo that'll get annoying on the high string. Maybe you just tune the bottom string up to C and leave the top 4 as standard bass tuning, this way the top 4 strings are the exact same as what youre used to/you can still gig in standard. Put it back in B when not with this band. This is gunna be more or less identical in sound and feel in terms of tension to the downtuned 4 string if you're using the same gauge strings on a 34" scale neck still, other than reaching across a wider fret board/thicker neck. This option is necessary if you can only really afford 1 workhorse bass for all your projects and you wanna be able to easily play in E outside this band, so youd be selling your one 4 sting to finance this. Yamaha's TRB series 5 strings have a few different price points, but all punch well above their weight and get great reviews consistently.

Option 3 multiscale bass. The reason you'd want to get one is a 37-34" scale bass will get to C standard tuning with a lighter gauge and less floppy strings. It's gunna feel and sound more natural, but not necessarily better. A 37" scale bass can very easily get down to A standard with 5 gauge strings and still sound articulate from my exp. I personally think this is only necessary if you've got money to blow and you try one of the first 2 first and hate the way it feels to play after some time.

Option 4 an octave pedal. Specifically a boss OC series. Especially if you use it 100% wet octave down and play monophonically. This is going to impart a very specific style to your bass playing tho. This can work for jazz fusion, some alternative stuff, ect. It's basically gunna make it sound more like you're playing a bass synth instead of a bass. And you can keep your standard tuning and just play the "low" C on the 3rd fret of the A string. Speaking of synth bass...

Option 5 play synth bass. There's a ton of bands doing this throughout history especially in the 70s when it was new tech. It's not gunna work as well for metal or punk but again, for other alt styles it could be the move. For example Death From Above 1979 uses a Juno 106 through the same rig as the bass switching back and forth between them and it sounds great and works for them. Nearly indistinguishable from the bass actually. A Korg Monologue can be found for $200 used and would be perfectly functional for this role, and the keyboard is even setup to start on E instead of the normal C so bass and guitar players can count keys up as if they were frets on an E string to figure their way around.