The register
The register lists Complexio Limited in London as a licensed sponsor on Skilled Worker, an entry that goes back to at least 2026. Its rating on the register is Worker (A rating).
At Companies House the business is Complexio Limited (company number 14463365), a private limited company incorporated in 2022, 4 years ago. The registered office is at First Floor, 5 Fleet Place, London, EC4M 7RD, EC4M 7RD. Its filed activity puts it in it, software and media. It has traded under other names before, including Quintessential AI Limited.
Holding a licence is the expensive, slow part done. For Complexio Limited the question each time is only whether the role and the pay fit the current rules, and that is checked per vacancy.