Yes, it is, but you will need to have an EK ticket office handle this.

Your existing booking has a PNR, which covers the 2 of you.

If you want to split this - and say travel on different days (same route), then what EK will do is keep your partner/colleague on the original PNR and then create a separate PNR for you for your outbound leg, and another PNR for your return leg.

I've only done this with EK on changing dates - but keeping the same route.(ie family of 4 all on the one PNR, then I had to change at last minute and return back to work early, EK gave me a new PNR for my return leg but kept the 3 other family members on the original PNR).

If you want to split the booking where you fly a completely different route, then I'm not convinced this will be possible, and if it is, it would probably be costly, with the re-faring of the new ticket and change fees associated.

If you can, do this face to face at a EK ticket office. If you try and do this over the phone, you may end up losing the will to live if you come across an agent who doesnt know what they're doing.