I guess I'm the high C specialist over here.
It depends a bit on the scale length of your bass and what kind of strings you like.
Often the high C strings don't fit well tensionwise to the normal set, so be prepared for some mixing.
Steel core strings are very thin and can cut into your finger, specially with higher tension. Synthetic strings and gut have a larger diameter but sustain might get short up the fingerboard, depending on the diameter. I recommend a diameter of 1 mm minimum (optimum). But this also depends a bit on tension.
Spirocore Mittel 4/4 works well on up to 105 cm 3/4 scale) instruments. On my 110 cm scale it was too much for me. There is an unofficial Spirocore 3/4 Weich high C that works well with the 3/4 Weich Spiros, at least for about one to two years before it gets a bit light compared to the G. But it's about the same tension as Spiro Mittel 4/4.
The Obligato high C works well with Spiro Mittel 4/4, but differs a bit in sound. Less good in high position than steel core.
I used a Efrano Medium high C (1.8 mm naked gut) with Evah Pirazzi Weich that worked well.
Unfortunately the Obligato G and Efrano Medium G have much less tension than the high C, so changing the G to the same set as the high C didn't work. I would prefer to have the G from the same set as the high C, but often the tension difference is worse than the sound difference to the G from the different set.
Personally I like a mix of some prototype strings from Innovation ("Studio") for high C, G and E with Innovation Braided D and A, because the "Studio" D and A had too less tension for my taste. Mixing the Studio high C and G with Spirocore Weich 4/4 D, A and E would be possible too. But I'm not sure if you can get the prototype strings from Innovation and I got three different high Cs from them, using only one.
The standard aluminium wound high C from Innovation has much less tension than the lower strings of the Braided set. I got a chromesteel prototype that was better, but still prefer the "Studio" high C over the Braided chromesteel high C.
There are also four high Cs from Corelli (I just tested two of them), several from Pirastro (I only tested Obligato), some from SuperSensitive (Supreme set discontinued, Pinnacle low tension, Sensicore aluminium wound and very low tension). Probably some more (like Pyramid Soloflex, only made on request).
Got mixed up now? Try one of the combinations of standard strings above based on the sets you like or try the complete Spiro Mittel 4/4 with high C if you think you can get used to the small diameter. Tuning down my 110 cm by a half tone was much better for me, which is basically the tension on pitch on a 105 cm scale.
Last warnings:
You might not be able to play high passages the way you are used to. Some passages are written to be played nicely on a G string, but you don't have the wide bowing angle on the G as before, so it's easy to accidentally touch the D or high C, specially if the string sinks either by a heavier bowing weight or simply by pressing the string down in a high position. Try it (with the D on your 4-string) and see how the string sinks down at the bowing point, making the bowing angle even smaller.
Not so much a problem in jazz and other improvised music where you can decide what and where you play and where you play more pizzicato than arco, but it might get difficult with music notated for the G as the highest string.
And when I changed from four to five with high C it took about a year to become used to the new tuning. And after that I sometimes got problems to use a four string, since I was used to have and constantly use the high C. So I have to use a four string bass every few weeks as well as the 5-string. I had problems missing the high C in jam sessions using other bassists 4-string instruments since my only instrument then was the high C 5-string. Currently I use a Stagg EUB as my 4-string for some practicing. The only other way would have been to take down the high C from my 5-string, but I didn't want to do that for various reasons.