The other application needs to support that behaviour, by defining a custom url scheme.
The developer needs add an URL Type i.e. "awesomeapp" in Xcode under Targets > Info > URL Types, if he does not support it you can not open the app.
In your Unity app just call
Application.OpenURL("awesomeapp://").
see also this stackoverflow entry:
Launch an app from within another (iPhone)
UPDATE 1: how to find a custom URL scheme of 3rd party app
- Download the purchased app via iTunes on OSX
- Go to the [AppName].ipa file on your filesystem
- Rename it to a [AppName].zip so that you can extract it
- Go to "[AppName]/Payload/[AppName].app"
- Right click and select "Show Package Contents"
- Open the Info.plist in Xcode and look for the key URL types or in a text editor for CFBundleURLTypes.
UPDATE 2: iOS 9
For iOS 9 you must whitelist that application in the Info.plist of the Xcode Project:
<key>LSApplicationQueriesSchemes</key>
<array>
<string>awesomeapp</string>
</array>
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