What you need to ask yourself is what will Microsoft Office 2016 do that you cannot do with Microsoft Office Pro 2003. Also remember Office 2016 will not have Access supplied, but your Office 2003 does. If you are not worried about Access and you want/need the new features provided in Office 2016 over Office 2003 then consider moving to Office 2016.
As to moving the new Office 2016 to a new computer, why not wait until you have purchased the new PC and install it there and leave the Office 2003 package on the old machine.
A lot of alternative Office type packages indicate that they are able to open Microsoft Office documents, but how well they do this may depend on how many of the lesser used features of Office that you rely on.