There are three basic ideas that should be considered when thinking about the characteristics that makes a great agile coach:

1. They are the embodiment of Agile Values and Principles. The Agile Coach has to be seen to living the values and principles, not just professionally but personally as well.
2. They are coaches, and so reflect general notions and professional values that all coaches have. This means that, for example, the Agile Coach takes an interest in developing their professional skills through organizations such as Agile Coaches Competencies via the Agile Coaching Institute.
3. They are able to work with Teams themselves, and so often start out as a Scrum Master. In fact, this could be considered a pre-requisite for a coach as, if you cannot lead a Team, you probably should not be a coach.

The last point tells us that many of the characteristics of a great Scrum Master will be needed if you wish to be a great Agile Coach. Referencing What Are The Characteristics of a Great Scrum Master?:

• Intimately familiar with Agile.
• Intensely curious.
• Responsible.
• Excellent facilitator.
• Collaborative.
• Influential.
• Highly available to the team.
• Be quiet.
• All about the Team.
• Sensitive “BS” detector.