Can I install 10.3 on my Mac?

No. You cannot install an OS earlier than what your Mac came shipped with. The earliest you would be able to run is Snow Leopard.

A brody is correct, HighSierra is the latest that you can run.

If you have a HDD inside, just upgrade.

If you have a SSD inside: HS will convert to APFS.

(there is a trick to avoid the APFS conversion).

If Sierra runs on your Mac, so will High Sierra 10.13. Just one caveat. SSD drives will be automatically formatted a new file system than Sierra and older systems can't see.

And no, your Mac is not compatible with Mojave 10.14, unless you use a hack.

Backup your data before any install.

High Sierra upgrading tip

Explains more.

Which model of iMac do you have?

I am thinking you mean 10.13 High Sierra (2017), not 10.3 Panther (2003).