Looking to moving to alicante to work

Confusion, sorry. You have a "European passport" by which you mean you are looking for work as a EU citizen.

In which case, a work visa is not required. EU citizens can live and work in another EU country without a visa but you do need to sign on the register of EU foreigners residing in Spain in order to stay more than 90 days.

I am a British citizen but I still have a valid EU passport. 

Then marcliff's post is good advice. 

Steve 

If you're looking for work as a UK citizen then, quite simply, you won't get one. You need a work visa which your employer needs to apply for so you need a job first. It must either be a/ A job that is on the shortage list such as high tech IT management, scientific research, engineering or some such. There is a list of what are classed as shortages.

Or b/ Your employer has to to show that no EU citizen could be employed in the job. 

Therefore, your "EU passport" (there isn't actually a EU passport, just a national one of a country in the EU) and I'm assuming Irish as you haven't said, means you can move to Spain and work without a visa although you'd need to sign on the foreigners' register. 

You would also need, probably, a good grasp of Spanish unless you are going to be working in an English pub or restaurant or a British supermarket etc although even the latter requires basic Spanish. Also very poorly paid and often without a contract and very seasonal.

I was talking to a dentist who had advertised for a receptionist. He picked a Polish lady and explained that every British applicant could only speak English but was "trying to learn Spanish". The Polish lady could speak Polish, Spanish, English and German.

Even the barmen and waiters at our local places have a fairly good smattering of several languages. 

Not wishing to put you off but you need to sort it out using whichever nationality suits you.

When you say European passport do you mean such as an irish passport or other eu state or a red U.K. passport which still says European Union on the front cover 

I have a red European Union passport 

I will get there eventually, I won't give up.

If it says United Kingdom of Great Britain and Northern Ireland then you do not have an EU passport as we are no longer in the EU.

As said, the only way to get a work permit is to have a job offer and your future employer must state that and you need to show a copy of the work contract stamped by the Foreigners' Office.

Good luck but UK is no longer an EU country despite the passport saying European Community. You are also restricted to spending 90 days in the EU in any 180 day period. 

Hopefully you didn't vote for Brexit?