Should we use a digital low pass or high pass filter to remove sensor bias?
If we use a discrete derivative with a discrete integral in series, will this remove (atleast some of) the DC bias in the signal?

Yes (if your integrator starts at state 0).
Note that this is a CIC filter.
Also notice an offset should be present in your accelerator data, or you just got lost in space.

Correspondingly it seems a better idea to use a high pass filter rather than a low pass filter although this will invite noise.

Well, seeing that by definition of &quot;low-pass filter&quot;, the DC component remains untouched, this doesn't only seem to be a good idea, but mandatory.

Is a notch filter how people deal with accelerometer signals and should one go with a high pass approach in case of the unavailability of a notch filter (since the signal necessarily contains bias)?

It depends on what people are doing for what purpose. My guess is you want to estimate something from your accelerometer observation. Maybe Kalman filters are what you're after?

Does the same logic hold if one designs the filter in the continuous domain?

This has nothing to do with continuous or discrete time. So, yes.