We study instabilities of the flow between two eccentric cylinders where the inner one is rotating, the outer one is fixed and a pressure gradient along the axis of the cylinders gives an axial flow (eccentric Taylor-Couette-Poiseuille flow) as an extension of studies on Taylor-Couette flow. The flow is characterised by three non-dimensional parameters: rotational and axial Reynolds numbers and eccentricity (offset of the cylinders' axes normalized gap width, or the difference of radii of the outer and inner cylinders). Due to these three independent parameters, it is expected that the flow becomes highly complex. Our aim is to understand the flow and to extract the essential physical parameters. We study the onset of instability through reflective-flake flow visualizations. For a given axial-flow Reynolds number and eccentricity, we carefully increased the rotational Reynolds number to determine the critical rotational Reynolds number for which the flow instability structure begins to appear. The results show that the observed critical points agree with linear stability analysis by Takeuchi and Jankowski (1981); Leclercq et al. (2013). Now, we re-scale these neutral/critical curves using new scalings: a modified Reynolds number based on the bulk azimuthal velocity and the local narrow gap width; and ratio of the axial bulk velocity to that in the azimuthal direction. The results clearly indicate that the instability of the whole flow is dominated by the narrow-gap flow, which is strongly influenced by the three-dimensionality of the mean flow quantified by the bulk velocity ratio. For the lower and higher bulk velocity ratio, the flow seems to be dominated by different instability mechanisms: centrifugal and shear instabilities, respectively. In the talk, we will also discuss these scalings in similar three-dimensional canonical flows, e.g., flow around a rotating cone in axial flow (the cylinder can be regarded as one limit of the cone, having an apex angle of zero degrees).