A fantasy bike on an unlimited budget??
Hmmmm...
The particular road bike I'd want already exists; I've even seen it first hand: a Mondonico Monza; black; chromed fork crown, stays, & HT lugs; internally ducted R. brake line; full Campy Daytona 10sp. I'd gladly settle for a Kalavinka Super Exhibition with DA 7700; DT wheelset.
But as far as would defined as something personal & unique, I guess I'd prefer a 650a or 700c touring bike, built along the lines of an early '60s Toei Stag, albeit with a full set of touring braze-ons for racks, panniers, etc. Alex Singer lugs; internal brake cable routing; internal generator cable routing (similar to my Toei); King dropouts; maybe pump pegs on the L. seat stay. The tubeset would be pretty unique, some Kaisei equivalent of Ishiwata EX or EXT, albeit with plain-guage chainstays- bent, like those of an early Stumpjumper; kickstand plate; Pacenti biplane fork crown. I'd be inclined to keep it all JIS (even if Chris King doesn't make JIS headsets).
Paint to match the 1985 Raleigh Kodiak, and built either by Greg Morris (formerly of Milholland Cycles, now with Vanilla) or Hiroshi Iimura (of Jitensha Studio, since this is in the style of a classic Toei). If I could reach back through history for any builder, I'd say it'd have to be Watanabe.
Gruppo could vary, but would likely be simple, just 8-speed with bar-end shifters; I'm partial to Deore XT (M737 group) but also have a soft spot for RSX/ RX100; Shimano 105/5500 would also work; . The crankset would be troublesome; part of me would inclined toward Sugino, the other part toward Shimano; triple in either case. Brakeset would be Paul Touring cantis; Phil Wood BB; Nitto stem, bars, & seatpost; Brooks saddle, likely a Flyer Special. Wheelset would be built around sealed-bearing hubs, 36h; fantasy says Maxi-car, but reality says White. Maybe Phil. Rims could vary a lot since there are a fair number of very reliable 700c touring rims out there.