There's an app for that

There are, in fact, several decimal time apps for the iPhone.  I found three which display French decimal clocks.

• French Revolutionary Clock by Miller Tinkerhess is simply a photo of an antique decimal clock from the French Revolution, with the hands animated to show the local time in decimal hours, minutes and seconds. There are no other features, but it looks pretty cool. It costs $0.99.  
  
• DeciTime by tin:b Software (Robert Fisher) uses colored arcs or slices to display local decimal time. The colors can be changed and there are four different analog clock styles, including one with regular hands. The units are called "dour", "dinute" and "decond", apparently after http://serg.us/dime/. The decimal time can also be displayed digitally. It also costs $0.99.
• MetricClock by ecce is also $0.99. I have not downloaded it, but it apparently differs from the others by allowing different background images to be used. Either analog or digital time can be displayed, and it also shows progress bars for the day, year and/or your expected lifetime.

If you prefer the time in Switzerland to your own local time, there's an app for that, too.

• Internet Beat by Thomas Cherry displays the Swatch Internet Beat time, which divides the day into 1000 "beats", from @0 to @999, with @0 being midnight at Swatch's HQ in Biel, Switzerland, i.e. Central European Time.  When you rotate the phone, it shows hundredths.  This is a free app.

There are some apps which display Unix time used by some computers and some which show date/time used by astronomers.

• Astrodate by Flying Titans, Inc., which converts the current date and time into several varieties of Julian Date. It costs $0.99. 
• iEphemeris Lite by Marco Piccone displays the current Julian Date, along with astronomical data on the moon, and is free.

There are a number of apps which use decimal time for payroll and such, which divide hours into hundredths, like employee time clocks do.

Finally, I also found a couple of stardate apps, which display a "stardate" based on the current date, although they each give very different results, and are not exactly what I would consider decimal time, but may be useful for trekkies who want to geek out. :)

•  iStardate by SusaSoftX ($0.99) displays stardates which look like [-28]1456.06.  I dunno, that doesn't look to me like the ones in the captains' logs.  Brackets?  Minus sign?  I know these are the same stardates used by Google, which were devised by Andrew Main, but I never liked them.  Anyway, the app is supposed to be used as a clock, which I guess means it increments 0.01 every 2 or 3 minutes, or 5.00 per day.  Meh.
• StarDate Calculator Pro by Triangle Powers Software is free.  (Why "pro" then?)  It opens with the current stardate, which looks like 63201.5, and you can select other calendar dates to convert, although for some reason the dial always starts on September 26.  These stardates increment about 2.7 per day, or 1000.0 per year.

MJD 55167.1