Light jet Captain reported following a false glide slope prior to capturing the localizer; which resulted in a low altitude alert from ATC while on a visual approach to SUS. Crew reestablished a stable approach and landed normally.

Narrative

On visual approach into SUS; I got distracted by what appeared to be a false glide slope indication; indicating I was too high. In following it; I got low enough to receive a low altitude alert by the tower. I was using the ILS to back up the visual approach; and I believe I started to descend before established on the localizer; just prior to intercepting. This; along with my apparent loss of situational awareness; meant descending prematurely. We were in visual conditions with the airport in sight. When I realized the Visual Glide Slope Indicator climb rapidly; now indicating we were below slope; and my sight picture outside beginning to appear too low; I climbed and received the low altitude alert. I finally saw the VASI and continued the approach since we were above 500 feet AGL and stabilizing. The aircraft did not give us any taws or other warnings. We debriefed the event as a crew post flight and I realized even though on a visual approach I should have not used the GS indication until established on localizer; and should have maintained better situational awareness of my distance from the field prior to descending. If I would have spent more time looking outside I probably would have seen the VASI sooner. I also could have cross checked with the FMS or MFD (Multi-function Flight Display) for better situational awareness of distance and appropriate altitudes.