The answer may not make sense to you

Question

I am

Half an idea

Fifth of a cm

Partly a reminder

Fully a doctor

And a desired statue

Who am I?

Answer 1

Might 'you' actually be:

ME?

Half an idea

ME is half of MEME. (Either half, in fact...)

Fifth of a cm

ME makes up 20% of the 10-letter word 'CENTIMETRE'.

Partly a reminder

ME makes up part of the words 'MEMO(RANDUM)', 'MEMORY' and 'MEMENTO'.

Fully a doctor

M.E. is an abbreviation for Medical Examiner.

And a desired statue

An 'Emmy' (say it out loud: "M.E."!) is an award for excellence (in the form of a statuette) desired by many in the television industry.

And as for the title, well, the answer may not make sense to 'you', but...

...it does to ME!