Leaving a company and becoming a consultant can be rewarding, both personally and financially. However, it is not necessarily simple.
First, consider whether consulting on the product breaks any Non-Disclosure Agreements (NDAs) or non-compete clauses in your contract. Whether these are enforceable or not does not help you when you are facing down lawyers. The customer may also have a clause in their contract stating they cannot hire anyone from your company or vice versa.
Second, that being a consultant is different to being a developer. You are using the product rather than writing it. Forget the nice IDEs you are used to and start thinking of batch files/scripts, database migrations and other less glamorous things.
Third, your knowledge of the product is going to atrophy over time. You will not have access to the source code and, as improvement and modifications occur, you will not know what they are until you use the product like everyone else. You may have friends on the development staff but do not expect them to give you the inside scoop on everything forever.
Lastly, being tied to a single customer can be dangerous. That customer may go out of business or stop using your product. You will need to start networking more and expanding your list of contacts. You may enjoy this or not but you have to be good at it to succeed.
[Edit: Added the following.]
That said, many people become consultants. Some people find development does not give them the same buzz and want to focus on the broader (and potentially more lucrative) customer problems. Some just have interests in many areas and consulting is a way of satisfying all of them. Assuming you are OK with the issues given above, this is a perfectly acceptable and reasonable route to take.