We need to develop quite a powerful web application for an investment bank. The bank IT would like us to build it on top of the SharePoint platform
UPDATE: After looking at this more I would add that I do not believe that SharePoint is for you. As I mention below SharePoint is for collaboration. If the users that come to the site require an isolated experience then SharePoint is more overhead than you need.
SharePoint is built on top of ASP.NET so you have everything that you want to do with ASP.NET in addition to what SharePoint provides. Anyone who says that it is difficult is trying to make it that way. You can deploy stand alone custom pages with 100% of your own code and it will run under sharepoint, or you can create new application pages that also contain any code you want to write, or you can simply add your own webparts that can be added to any page you choose with 100% of your own code.
Here is just one example.
Creating an Application Page in Windows SharePoint Services 3.0
What SharePoint offers on top of that is a whole different paradigm on collaboration tools. If you wish to leverage it (if not the cost on return is somewhat limited) you can build amazingly complex and integrated solutions that is build around the aggregation of data from across an enterprise.
That being said, do not go into it lightly. If deployed wrong or with a half understanding of where SharePoint excels and where it does not will result in a diaster. Unless you have the time to understand the core concepts of SharePoint I would warn against it but your client is right. If you do build it in SharePoint you get a great deal more flexibility. One right off the bat is the ability to mix authentication modes. I designed a solution that mixed custom forms authentication with an LDAP backend with Windows Authentication. Anyone could visit the same pages but your authenticated account could come from two different locations.