You won't be able to do this exactly as you describe without overriding your standard layout with a Visualforce page. But there are some alternatives that will provide a visual indicator of the status without having to completely override the page.
First, you can use a formula field with a CASE() function that displays a different image depending on the picklist value. You can put the formula field above, below or next to the CaseNumber field so they are near each other. The formula will look something like this:
IMAGE(
CASE(
Status,
"New", "/path/to/image/file",
"Closed", "/path/to/image/file",
"Escalated", "/path/to/image/file",
"/path/to/image/file"), /* default */
"Status"
)
Here are some links for more information about images in formulas:
Option two would be to create a Visualforce page with the Case standard controller, that sets the background color of a div based on the Status field value. There doesn't have to be anything in the div, though you could display the CaseNumber value in a contrasting color. Then add a single-column section with no header at the top of your page layout, and add the VF page to that section, setting the height to 20-30 pixels. You'll end up with a colored bar along the top of your page which will convey the status.
The Visualforce would look something like this:
<apex:page standardController="Case" showHeader="false" standardStylesheets="false" sidebar="false">
<div style="background: {! CASE(Case.Status, 'New', '#f00', 'Closed', '#0f0', '00f') }; text-align: center;">
<apex:outputField value="{!Case.CaseNumber}" />
</div>
</apex:page>