The current clamps for scopes are low value resistors with the correct gain setting wired into the plug. Nothing fancy, you can get ones with current sensing devices, but these themselves are, wait for it, low value resistors and guess what, the voltage developed across the resistor is used to represent the current. If you want to spend lots of money you can get magnetic sensors that use the Hall Effect to detect current, but guess what, these also detect a differential voltage and apply the correct gain. If you want to spend even more money, you can use a tuned inductive corner current sensing device, but, this uses the Hall Effect and you've guessed it this uses a differential voltage to represent the current, damn you just can't sense current without voltages getting involved. Well you can, but this adds an extra load, (usually complex rather than a very low impact purely resistive load), which is not really what you want, all those extra calculations, and not to mention the circuit would not behave in the same way, so all those extra calculations would be complex as well.
You can't really win with this one. The cheapest and perhaps one of the best ways is to use a low value resistor in series and apply the correct gain or vertical setting on the scope. This will show you the current waveform in the time domain. If you do not want to interfere with the current and just sense it, this is the best way without spending lots of money.