I have deleted the default admin account many times. First I needed to create a new admin account and then just login with the new admin and remove the old one. As ivanreis said please make sure to re-assign all the knowledge objects of admin to the new user otherwise they will become orphan and you won't be able to use them.
I never experienced this customer request before. You have the option to clone the admin account to a new account. Also is important to highlight that all knowledge objects already created under admin account have to reassigned to this new user.
There is an old topic at splunk answer about the same topic, although it is related with the older Splunk versions, can be applied to your case as well.
In this case, my suggestion is to run a new Splunk enterprise installation and when splunk will start for the 1st time, you have to type the new admin user requested by customer, check this doc