In case it's not clear in the above comment a magnetic field arises from the movement of charge--it doesn't matter if you move past the charge or the charge moves past you.

Suppose, we put a charge on a train, and have two sets of electric and magnetic field detectors. Put one set of detectors on the train next to charge, and another on the platform of the station.

When the train is sitting at the station, both detectors will measure an electric field and no magnetic field due to the charge.

Once the train starts moving, the detector on the train will still measure an electric field, but no magnetic field. The detector at the station will measure both an electric and magnetic field.

Repeat the experiment, but this time put the charge on the platform. You get the same result except it is the train detector which will measure both the electric and magnetic field while the detector at the station will measure only an electric field.

Whether or not you measure the magnetic field is dependent upon your reference frame. This was one of the ideas which inspired special relativity.