Do we have in practice any analog filter that has a complex impulse response? Or they must have a real impulse response? Particularly, I am looking for a low pass filter that passes only a band in positive frequencies.
Complex impulse response makes no sense. You put a blip into a system, and something comes out. While complex math may be useful for modeling a system and performing operations on that model, it doesn't change the fact that any real system always has some real response.
What would it even mean for the result of a blip to be complex? What would you expect to see on a scope if the response was complex somehow? The real system does something when you put a blip in. That something exists in the real world.
Complex voltage? That does not exist. Even in theoretical discussions it actually is 2 normal voltages. One of them is the real part and another is the imaginary part.
In modulation and demodulation theories we usually have complex sinusoidal voltage. There Real and imaginary parts are sine voltages which have 90 degrees phase shift. That unifies gain and making phase shift. Both can be achieved conveniently by multiplying with complex number.
In many modulation, demodulation and also phase locked loop schemes we see 2 parallel identical signal paths - one for the real part, one for the imaginary part. The first is normally said to be the I channel and the latter is the Q channel. Often they are only program in a signal processor, but I have had in my hands also a real analog circuit with I and Q channels. There were also some filters. Their simultaneous impulse responses together can well be considered as one complex impulse response assuming a proper complex impulse is inputted.
Beyond The Invisible: You probably at least dream of something where complex physical quantities are measurable and meaningful. But our current reality has only usual voltages which have no imaginary part. Those voltages have both the positive and negative frequency components at the same time in their spectrum.
By having 2 separate sinusoidal voltages with 90 degrees phase shift and 2 filtering circuits, you can make the difference between positive and negative frequencies. In one wire they have no difference.
What about non sinusoidal complex signals? For example how to generate complex, but still absolute unprocessed speech from a mic output to be filtered? That's nonsense until you have a person who has real and imaginary parts in the output of his mouth and a complex output mic which catches them.
While there are no analogue complex filters in the sense I think you mean, there are allpass networks that are used to build SSB demodulators given baseband I and Q, the networks are component heavy, sensitive to tolerance and generally a bit of a pain, but they do work.
Here for example is one paper on the subject, http://yu1lm.qrpradio.com/AF%20ALL-PASS%20NETWORK-YU1LM.pdf