Physical understanding of vortex flow instability is one of the essential prerequisites for proper prediction and analysis of the behavior of vortex flow encountered in many engineering problems. In this study, a stability analysis code for axisymmetric vortex flow is developed based on the linear stability theory. A stability equation is derived from the governing equations for general compressible flows in the cylindrical coordinate system in order to take into account both the effects of compressibility and viscosity on stability results. The code is validated through analyses on widely-known model vortex problems by comparing the results with those from preceding researches. The comparison of the results confirms that variations of unstable eigenvalues with respect to vortex and disturbance parameters as well as the maximum amplification rate can be obtainable very accurately from the developed code. The effect of viscosity on the instability characteristics of supersonic vortex flow was further analyzed by carrying out the additional analysis. The results show that the analysis of the present study provide a more accurate prediction on linear amplification of disturbances compared to the inviscid analyses carried out in former studies. This also indicates that the present code can be considerably useful for parametric studies on the instability of compressible vortex flows with viscous effects.