There will junctions along the way - defo at Newhaven, Rottingdean, Marina, Lagoon, Worthing etc. you get the idea ....
Bit like the M1 London to Leeds - not many people drive end to end, but hop on and off to suit.
Will probably need to be a toll road like M6 toll to pay for sea damage too.
I heard someone talk about building a 6 lane [3 each way] motorway from Dover to Plymouth 100 metres out to sea.
Would solve all the south coast traffic congestion issues but might be a bit of an eyesore?
Bit like the road from Miami to Key West kinda thing.