If you mean by the same job, the same quality of cutting, then yes. A fixed base is supposed to be somewhat more accurate and less top heavy than a plunge router, but the accuracy hasn't been that apparent to me. If you mean can they both be used for the same things/applications then no. A plunge router can be more useful in certain applications where moving the depth during the cut is needed.
Oboy, that would be asking for trouble, Bill. Not that it couldn't be done, but it'd be awfully hard to get it done using nothing but hand control to make the depth variations.
A plunge router really shines over a fixed-base router when it comes to "plunge" routing - this means that the bit is held above the wood before you turn the power on (but the machine itself is resting on the wood), then with the power on you "plunge" the bit into the wood to a preset depth. You can then slide the router sideways if necessary with the bit still "plunged" into your wood, say to rout the edge of a panel or to cut a sliding dovetail or a keyhole slot.
A fixed-base router, on the other hand, really should be used only for work which allows you to start the router with the bit already extended - like for the edges of panels, or for hinge mortises, or for dovetails with a dovetail jig.
I've been known to use a fixed-base router as a plunge router under certain circumstances, but I would NOT recommend that practice to anybody who isn't thoroughly familiar with routers and what can happen if you do it wrong!
As is the case with all tools, it depends on your primary use. I never buy a tool until I have a specific need and then I look at other uses as part of the selection process.
If you are going to own only one router, get a plunge router. If you plan to own two, I would go ahead and get a plunge router for hand routing and then consider a fixed router mounted in a router table whose bit height you adjust using one of the many incredibly cool height adjusting add-ons for router tables.
One thing I've read is that fixed routers are typically built using a beefier motor/shaft assembly since they are designed to be stationary. Plunge routers on the other hand are designed with a lighter motor/shaft assembly due to the "portability" requirements of the plunge design. I'm not sure if this is still the case with the number of combo units available. Perhaps the manufacturer's of Fixed "only" routers are building them this way.
Personally I chose a combo unit from Makita which I got for a steal of a deal a few weeks ago at a local woodworking show. I've also heard/read rave reviews on the new Bosch units but these units weren't as comfortable for me as the Makita's were during operation.
The Makita's were rated as a pretty close 2nd to the Bosch unit. Upon inspection the differences (for my needs) were nominal.