Yes. Clifford Algebras are beyond what I ever studied, so I have only a hobby knowledge about them, but I think all these hypercomplex numbers are Clifford Algebras - and many of them turn up as useful tools in the real world. In this group, many will be familiar with using complex numbers for filters, signal analysis, impedance calculations, etc. Some might have used quaternions for 3D work such as rotations in graphics, or for g-sensors, IMU's and navigation.
Extensions of real (or complex) numbers to include an "infinity", or multiple infinities/infinitesimals have plenty of uses. Adding a single "infinity" to a two dimensional real space gives you a sphere. Infinities and infinitesimals can be helpful when dealing with functions that have discontinuities or jumps - they can be useful to prove results which are then used blindly by engineers. Engineers are happy to take the Fourier transform of a step function, or play with the Dirac delta function - it is mathematicians using infinities and infinitesimals that prove what works with this kind of thing. (I don't know if surreals are used with these examples - that's beyond my knowledge.)
And if you are a TeX or LaTeX nerd, you might know that a lot of the lengths and other quantities are build from different levels of infinities and infinitesimals as well as real numbers.