Can quaternions be converted into positional data

Question

I have begun a basic incursion into understanding quaternions. I have read the following: https://en.wikipedia.org/wiki/Quaternion and https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation. I understand that quaternions are used to identify the rotation of an object in 3-dimensional space. I have seen quaternions be converted into 3D space via turning the Quaternion into Euler rotations but from my understanding, these are not positional coordinates rather another representation of rotation. I am wondering if it is possible to use quaternions to plot an objects path or if I would need something separate. I am asking in the context of IMU data with 9df (Accelerometer, Magnetometer, Gyroscope). I have applied the Madgwick filter to this data and received quaternion output. However, I do not believe I can use this alone to plot a path. As I do not have a Mathematics background I was hoping to hear opinions from people with greater expertise.